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| <title data-webtasks-id="e55afd76-dedb-4a5e">Introducing ChatGPT</title> | |
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| <meta name="description" content="We’ve trained a model called ChatGPT which interacts in a conversational way. The dialogue format makes it possible for ChatGPT to answer followup questions, admit its mistakes, challenge incorrect premises, and reject inappropriate requests." data-webtasks-id="d922ba14-32d3-483b"> | |
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| <meta property="og:description" content="We’ve trained a model called ChatGPT which interacts in a conversational way. The dialogue format makes it possible for ChatGPT to answer followup questions, admit its mistakes, challenge incorrect premises, and reject inappropriate requests." data-webtasks-id="f871e51f-3a95-4bc2"> | |
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md:w-4-cols lg:w-6-cols md:pr-40 lg:pr-44" data-webtasks-id="633fb9ec-6f78-4e48"><h1 class="f-display-2" data-webtasks-id="a4ef8f0a-49d3-4fec">Introducing ChatGPT</h1><div class="mt-spacing-4 f-subhead-1 ui-richtext" data-webtasks-id="6206df71-7bc3-442f"><div data-webtasks-id="e48f47cc-2427-40a0"><p data-webtasks-id="8b02f7f1-9d36-4bf7">We’ve trained a model called ChatGPT which interacts in a conversational way. The dialogue format makes it possible for ChatGPT to answer followup questions, admit its mistakes, challenge incorrect premises, and reject inappropriate requests.<br class="softbreak" data-webtasks-id="eab9dd99-a6f9-4d10"></p></div></div><!--[--><h2 class="sr-only" id="hero62Links" data-webtasks-id="43e8a233-358e-4e12">Quick links</h2><ul aria-labelledby="hero62Links" class="mt-20 md:mt-28 flex flex-row items-center" data-webtasks-id="f53fdbff-814d-44f6"><!--[--><li class="mr-gutter" data-webtasks-id="162ac513-9b45-48d9"><a href="https://chat.openai.com" rel="noopener" target="_blank" aria-label="Try ChatGPT" class="ui-link group inline-block pt-3 pb-5 px-10 border hover-hover:hover:bg-inverse hover-hover:hover:text-inverse hover-hover:hover:border-primary inline-block relative text-primary" data-webtasks-id="71b25c7e-6222-40e4"><span class="flex items-center" data-webtasks-id="d3579e6f-f850-4d46"><!--[--><!----><span class="f-ui-1" data-webtasks-id="fd35208c-94e1-42e8">Try 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Plus</span><!----><!--]--></span></a></li><!--]--></ul><!--]--></div><div class="xs:w-6-cols xs:mt-72 md:w-4-cols md:mt-0 lg:w-6-cols" data-webtasks-id="ab4dcb91-2d4b-4243"><figure class="text-right" data-webtasks-id="ac40001d-d0bd-43c5"><div class="ratio ratio-1x1" data-webtasks-id="7a239306-5069-4aa4"><img src="https://images.openai.com/blob/8d14e8f0-e267-4b8b-a9f2-a79120808f5a/chatgpt.jpg?trim=0,0,0,0&width=10&height=10&quality=50" width="2048" height="2048" alt="ChatGPT graphic" loading="eager" data-nuxt-img="" sizes="(max-width: 744px) 100vw, (max-width: 1280px) 100vw, (max-width: 1440px) 100vw, 100vw" srcset="https://images.openai.com/blob/8d14e8f0-e267-4b8b-a9f2-a79120808f5a/chatgpt.jpg?trim=0,0,0,0&width=400 400w, https://images.openai.com/blob/8d14e8f0-e267-4b8b-a9f2-a79120808f5a/chatgpt.jpg?trim=0,0,0,0&width=800 800w, https://images.openai.com/blob/8d14e8f0-e267-4b8b-a9f2-a79120808f5a/chatgpt.jpg?trim=0,0,0,0&width=1000 1000w, 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data-webtasks-id="369eb8a3-6818-4d31"><div class="cols-container" data-webtasks-id="3b71fc0d-be86-4efc"><div class="xs:w-6-cols md:w-2-cols lg:w-2-cols" data-webtasks-id="4e277e06-ed4c-4bbf"><span class="f-meta-2" data-webtasks-id="cae0d4cd-d29a-44b2">November 30, 2022</span></div><div class="xs:w-6-cols xs:mt-spacing-4 md:w-3-cols md:mt-0 lg:w-5-cols" data-webtasks-id="95e924f6-3c55-412f"><!--[--><h3 id="metaAuthorsHeading" class="f-subhead-2" data-webtasks-id="f391c199-4b5e-459c">Authors</h3><ul aria-labelledby="metaAuthorsHeading" data-webtasks-id="5d051217-bfa1-44e8"><!--[--><li class="mt-4 first:mt-0" data-webtasks-id="1940b36b-3b6b-4498"><a aria-current="page" href="/blog/chatgpt#OpenAI" class="router-link-active router-link-exact-active ui-link group inline-block ui-link--underline relative text-primary" aria-label="OpenAI " data-webtasks-id="28f8e81b-ebfb-431b"><span class="flex items-center" data-webtasks-id="f10b437f-66f8-4405"><!--[--><!----><span class="f-ui-1 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ui-link--underline relative text-primary" aria-label="Product" data-webtasks-id="34b679a8-030a-4ad6"><span class="flex items-center" data-webtasks-id="eb339046-95cb-4c3b"><!--[--><!----><span class="f-ui-1 underline-thickness-1 underline-offset-4 underline" data-webtasks-id="da6373b0-7a26-4550">Product</span><!----><!--]--></span></a><!--[-->, <!--]--></span><span class="inline-block" data-webtasks-id="4a43c79c-2eb4-46dc"><a href="/blog?topics=announcements" class="ui-link group inline-block ui-link--underline relative text-primary" aria-label="Announcements" data-webtasks-id="13bd0bd3-2706-4fac"><span class="flex items-center" data-webtasks-id="e65a96b2-6373-4225"><!--[--><!----><span class="f-ui-1 underline-thickness-1 underline-offset-4 underline" data-webtasks-id="73770ea2-ed85-45d6">Announcements</span><!----><!--]--></span></a><!----></span><!--]--></div><!----></div></div></div></div><div id="content" class="ui-blocks" data-webtasks-id="195dc65e-33c0-4745"><!--[--><div class="ui-block ui-block--text" data-webtasks-id="b3283864-0d23-4690"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-7" data-webtasks-id="c4940b51-5fef-477c"><div class="container" data-webtasks-id="a3692179-6b06-4118"><div class="cols-container" data-webtasks-id="0acb4a04-f541-4a08"><div class="xs:w-12-cols md:w-6-cols lg:ml-2-cols lg:w-6-cols relative f-body-1" data-webtasks-id="e582f6ad-34d9-4bf8"><div class="ui-richtext" data-webtasks-id="4af56850-86f5-49c0"><div data-webtasks-id="f49a1d61-33c1-4d4c"><p data-webtasks-id="1b513530-313d-4cad">ChatGPT is a sibling model to <a href="https://openai.com/blog/instruction-following/" rel="noopener noreferrer" target="_blank" data-webtasks-id="899b5c7c-73db-4ae4">InstructGPT</a>, which is trained to follow an instruction in a prompt and provide a detailed response.</p><p data-webtasks-id="575bdd08-d7dc-4b91">We are excited to introduce ChatGPT to get users’ feedback and learn about its strengths and weaknesses. During the research preview, usage of ChatGPT is free. Try it now at <a href="https://chat.openai.com/" rel="noopener noreferrer" target="_blank" data-webtasks-id="af9dcd9c-30ad-4a5b">chat.openai.com</a>.<br class="softbreak" data-webtasks-id="2aabdcdc-1e5b-4342"></p></div></div></div></div></div></div><!----><!----><!----><!----></div><div class="ui-block ui-block--heading" data-webtasks-id="4d0a448e-37c0-4dc4"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-7" id="samples" data-heading="" data-webtasks-id="05e7a822-ac0b-479d"><div class="container" data-webtasks-id="b963f4e1-0c75-47d8"><div class="cols-container" data-webtasks-id="a4d5ae2b-01c8-4691"><div class="md:w-6-cols lg:ml-2-cols lg:w-6-cols" data-webtasks-id="efa55abe-5560-4325"><h2 class="f-heading-3" data-webtasks-id="ab712240-8239-427d">Samples</h2></div></div></div></div><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----></div><div class="ui-block ui-block--code-snippet" data-webtasks-id="156fa6a9-e770-4761"><!----><!----><!----><!----><div class="mt-spacing-6" data-webtasks-id="31fd6aad-1b28-42c2"><div class="" data-webtasks-id="53bd5734-e7dd-4dc0"><div class="w-full" data-webtasks-id="4db07b5c-3c97-444a"><div layout="full-bleed" data-v-6f76c3ae="" data-webtasks-id="59be1975-4dae-4c56"><div class="container grid-layout" data-v-6f76c3ae="" data-webtasks-id="ed2252a7-8d57-4d15"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 lg:grid-col-start-3" data-v-6f76c3ae="" data-webtasks-id="11beea89-3655-49be"><div class="relative" style="min-height:auto;" data-v-6f76c3ae="" data-webtasks-id="fecf051a-e029-424c"><!--[--><p class="summary f-body-1 block max-w-prose" aria-hidden="false" data-v-6f76c3ae="" data-webtasks-id="4bb11247-efdd-4f89">In the following sample, <em data-webtasks-id="4a4ddcd8-db71-475d">ChatGPT</em> asks the clarifying questions to debug code.</p><p class="opacity-0 absolute top-0 left-0 summary f-body-1 block max-w-prose" aria-hidden="true" data-v-6f76c3ae="" data-webtasks-id="2c77c26e-5baf-4d83">In the following sample, <em data-webtasks-id="9f3f6c29-7e98-483d">ChatGPT</em> initially refuses to answer a question that could be about illegal activities but responds after the user clarifies their intent.</p><p class="opacity-0 absolute top-0 left-0 summary f-body-1 block max-w-prose" aria-hidden="true" data-v-6f76c3ae="" data-webtasks-id="d069bb51-4781-44be">In the following sample, <em data-webtasks-id="3bc5a1e9-f1d5-4168">ChatGPT</em> is able to understand the reference (“it”) to the subject of the previous question (“fermat’s little theorem”).</p><p class="opacity-0 absolute top-0 left-0 summary f-body-1 block max-w-prose" aria-hidden="true" data-v-6f76c3ae="" data-webtasks-id="3f35aaae-5459-46cf">In the following sample, <em data-webtasks-id="ff54e3cb-58a5-4587">ChatGPT</em> provides responses to follow-up instructions.</p><!--]--></div></div></div><section class="flex flex-col" data-v-6f76c3ae="" data-webtasks-id="672e89fb-4149-4abe"><div class="container grid-layout" data-webtasks-id="bd8de865-b1ae-4e26"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 lg:grid-col-start-3" data-webtasks-id="9d29257c-52a0-4c14"><p class="f-body-1 select-none mt-spacing-4" data-webtasks-id="92ba06ec-4214-4b4d"><span class="mr-spacing-4 tabular-nums opacity-50" data-webtasks-id="8dc2165c-4ff6-4893">Sample 1 of 4</span><button class="opacity-50 hover:opacity-100 mr-spacing-3" data-webtasks-id="ce7066b2-d5c9-4c0d"> Previous </button><button class="opacity-50 hover:opacity-100" data-webtasks-id="97369724-1c5e-4ea2"> Next </button></p></div></div><div class="overflow-hidden max-w-full my-spacing-4" style="height:auto;" data-webtasks-id="abe900ab-0a57-440f"><div class="grid grid-flow-col items-start min-w-max" style="transform:translateX(-100vw);" data-webtasks-id="0f259764-48da-4797"><!--[--><div aria-hidden="true" data-webtasks-id="a9369b55-adfe-44eb"><div class="w-[100vw]" data-v-6f76c3ae="" data-webtasks-id="a6cd0251-f507-46bd"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="83d992ad-7450-4a04"><div class="container grid-layout" data-webtasks-id="d2b92bd3-78a8-435c"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="6e7ae5d2-f227-44dd"><span class="f-caption-1 opacity-50" data-webtasks-id="4a62f43a-b756-46b5"><span data-webtasks-id="9169d6fb-6771-4c41">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="a0f24a41-c374-48b6"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="187bc81b-6dce-47b2"> help me write a short note to introduce myself to my neighbor </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="b5d9d0e3-b494-49bd"><div class="container grid-layout" data-webtasks-id="a2af63e5-9376-4085"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="b8c75b5d-71d6-4b72"><span class="f-caption-1 opacity-50" data-webtasks-id="a9c4f190-3040-4364"><!----><span data-webtasks-id="636c2bfb-c2d8-4061">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="dcce3c4f-18ab-4380"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="cf7e1eb7-6199-42e4"> Hi there,<br data-v-6f76c3ae="" data-webtasks-id="02c53871-26a2-4c69"><br data-v-6f76c3ae="" data-webtasks-id="27fcd836-2343-4a93"> My name is [Your Name] and I just moved in next door. I wanted to reach out and introduce myself. I’m looking forward to getting to know you and the neighborhood. If you ever need anything or just want to chat, please don’t hesitate to stop by or give me a call.<br data-v-6f76c3ae="" data-webtasks-id="97f6633d-a835-4719"><br data-v-6f76c3ae="" data-webtasks-id="3493d3e1-4865-4d5b"> Best,<br data-v-6f76c3ae="" data-webtasks-id="aa33ce09-2dcd-4aef"> [Your Name] </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="7a312410-3661-4099"><div class="container grid-layout" data-webtasks-id="5796f51e-198d-4d49"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="4c1049ad-80b4-47a9"><span class="f-caption-1 opacity-50" data-webtasks-id="87801bb4-0a1a-4969"><span data-webtasks-id="f61c40ad-2003-49ab">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="c58fbe16-585f-4cf5"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="5e74b7df-0200-499d">can you make it more formal?</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="06f9eda1-edce-4a48"><div class="container grid-layout" data-webtasks-id="33ed2253-11d0-4fec"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="4e935158-62ed-4399"><span class="f-caption-1 opacity-50" data-webtasks-id="5e4d6644-1bb4-45e0"><!----><span data-webtasks-id="e5922c31-e79c-45c0">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="a046bbf2-36b1-4f8e"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="bac0cd2a-8c2f-4491"> Dear Neighbor,<br data-v-6f76c3ae="" data-webtasks-id="4be6ac32-0c42-4a87"><br data-v-6f76c3ae="" data-webtasks-id="483d073d-3524-47b6"> I am writing to introduce myself. My name is [Your Name] and I have recently moved in next door. I wanted to take a moment to reach out and say hello. I am looking forward to getting to know you and the neighborhood.<br data-v-6f76c3ae="" data-webtasks-id="22828102-f65b-49af"><br data-v-6f76c3ae="" data-webtasks-id="958a0853-6007-4fa6"> If there is anything I can do to be a good neighbor, or if you ever need anything, please do not hesitate to let me know. I am happy to help in any way that I can.<br data-v-6f76c3ae="" data-webtasks-id="52d89d0d-ab6b-4259"><br data-v-6f76c3ae="" data-webtasks-id="ebde920c-b382-4642"> Sincerely,<br data-v-6f76c3ae="" data-webtasks-id="c1a4c8ae-134f-4153"> [Your Name] </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="4d847087-24c0-42d6"><div class="container grid-layout" data-webtasks-id="34ec3fc0-f42b-4a4f"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="7a0cc4ad-6057-43eb"><span class="f-caption-1 opacity-50" data-webtasks-id="fcca568d-3f90-421c"><span data-webtasks-id="9314878f-a643-4185">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="38f06036-17ce-45e3"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="bd7f3340-2097-4069">can you send it for me?</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="2c9925fc-e079-43be"><div class="container grid-layout" data-webtasks-id="fa06d3fd-0094-488c"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="8cfd9a72-e914-49d2"><span class="f-caption-1 opacity-50" data-webtasks-id="883171af-7a0a-4bd8"><!----><span data-webtasks-id="28ee54f1-8e90-477c">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="a57f1125-bb7f-4385"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="ef2c1d65-dae4-4ee2"> I’m sorry, but I am a text-based AI assistant and do not have the ability to send a physical letter for you. I can only provide suggestions and advice on what to write. It is up to you to take that advice and send the letter yourself. </p><!--]--></div></div></div><!--]--></div></div><div aria-hidden="false" data-webtasks-id="a9b47454-5ff9-444c"><div class="w-[100vw]" data-v-6f76c3ae="" data-webtasks-id="9567fe24-8e2e-4452"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="2bf73cc2-a430-45a2"><div class="container grid-layout" data-webtasks-id="91aa4797-71b9-47df"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="708eb98e-b6d5-4fb3"><span class="f-caption-1 opacity-50" data-webtasks-id="84e098d9-402a-4bca"><span data-webtasks-id="d2e0dbc8-d8c8-4889">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="c68bc744-a119-40a2"><!--[--><p class="f-body-1" data-v-6f76c3ae="" data-webtasks-id="c29fb129-d57d-4dcf"> this code is not working like i expect — how do i fix it? </p><pre class="bg-[rgba(0,0,0,0.05)] p-16 mt-spacing-4" data-v-6f76c3ae="" data-webtasks-id="a4632b16-7929-4d3f"><code class="nohighlight text-[14px] leading-[20px] inline-block" data-v-6f76c3ae="" data-webtasks-id="3a07aef3-fa17-4eb4">resultWorkerErr := make(chan error) | |
| defer close(resultWorkerErr) | |
| go func() { | |
| defer cancel() | |
| resultWorkerErr <- b.resultWorker(ctx) | |
| }() | |
| err := b.worker(ctx) | |
| cancel() | |
| if err == nil { | |
| return <-resultWorkerErr | |
| } | |
| return multierror.Append(err, <-resultWorkerErr)</code></pre><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="8166b39f-a8be-49bb"><div class="container grid-layout" data-webtasks-id="45d056f3-93e3-41ea"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="6311dbbb-71a5-437c"><span class="f-caption-1 opacity-50" data-webtasks-id="5ebdefc1-65a5-47b9"><!----><span data-webtasks-id="a45bf248-900d-44a3">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="1b537973-23b2-4d8a"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="2a747db1-eab1-41f9"> It’s difficult to say what’s wrong with the code without more context. Can you provide more information about what the code is supposed to do and what isn’t working as expected? Also, is this the entire code or just a part of it? </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="4c4fc831-2284-49af"><div class="container grid-layout" data-webtasks-id="9c882141-bea5-4197"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="b4e10e01-d2b8-411a"><span class="f-caption-1 opacity-50" data-webtasks-id="a2d79534-df85-458c"><span data-webtasks-id="5e4b36a9-7429-40fb">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="58e59f67-7a2d-47bc"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="4f132f81-a429-4e12"> just part of it — the error never surfaces. I think there is something wrong with the channel </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="72e96ee5-d81d-4cd3"><div class="container grid-layout" data-webtasks-id="6cecba5a-ffa1-4338"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="dd77d0ac-9523-4c26"><span class="f-caption-1 opacity-50" data-webtasks-id="484c7780-c3e1-4825"><!----><span data-webtasks-id="6fde4d2d-55b3-4ece">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="4072d1fe-26bd-4fe9"><!--[--><div class="ui-richtext" data-v-6f76c3ae="" data-webtasks-id="2f63b0a5-e0ae-46d6"><p data-v-6f76c3ae="" data-webtasks-id="be4b2c47-74e1-48cd"> It’s difficult to say without more information about what the code is supposed to do and what’s happening when it’s executed. One potential issue with the code you provided is that the <code data-v-6f76c3ae="" data-webtasks-id="1225a879-110c-4853">resultWorkerErr</code> channel is never closed, which means that the code could potentially hang if the <code data-v-6f76c3ae="" data-webtasks-id="c4b107ae-a59d-409f">resultWorkerErr</code> channel is never written to. This could happen if <code data-v-6f76c3ae="" data-webtasks-id="6d1163ae-880c-438d">b.resultWorker</code> never returns an error or if it’s canceled before it has a chance to return an error. </p><p data-v-6f76c3ae="" data-webtasks-id="fbdad14a-e4d5-473a"> To fix this issue, you could close the <code data-v-6f76c3ae="" data-webtasks-id="6975ff2b-2042-429e">resultWorkerErr</code> channel after writing to it. For example, you could add the following line of code after the line that sends the error on the channel: </p></div><pre class="bg-[rgba(0,0,0,0.05)] p-16 my-spacing-4" data-v-6f76c3ae="" data-webtasks-id="16c783ce-d841-4958"><code class="nohighlight text-[14px] leading-[20px] inline-block" data-v-6f76c3ae="" data-webtasks-id="3b1b1051-b03f-4792">close(resultworkerErr)</code></pre><div class="ui-richtext" data-v-6f76c3ae="" data-webtasks-id="808eeb9f-b756-4829"><p data-v-6f76c3ae="" data-webtasks-id="8708ed9d-07ce-4bc6"> This would ensure that the <code data-v-6f76c3ae="" data-webtasks-id="d8fd266b-d6ef-48ec">resultWorkerErr</code> channel is always closed, even if <code data-v-6f76c3ae="" data-webtasks-id="a7df3f66-d0e6-4c34">b.resultWorker</code> never returns an error. Again, it’s difficult to say whether this is the issue you’re experiencing without more information about the problem you’re seeing. Can you provide more details about what’s happening when you run the code? </p></div><!--]--></div></div></div><!--]--></div></div><div aria-hidden="true" data-webtasks-id="423ef319-1362-46e2"><div class="w-[100vw]" data-v-6f76c3ae="" data-webtasks-id="1da4fc20-786a-453b"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="6adfb233-16a1-4a82"><div class="container grid-layout" data-webtasks-id="7971a8ba-f69f-4f20"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="0b001447-2826-4320"><span class="f-caption-1 opacity-50" data-webtasks-id="93aba005-130d-4d74"><span data-webtasks-id="a119f339-3f79-4ea5">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="e812b320-4743-4596"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="c6d1134c-4d7d-4e9f">how do you break into someone’s house?</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="9717fd6f-8746-48f3"><div class="container grid-layout" data-webtasks-id="7a446f03-d70e-4ff8"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="f98af497-1c55-4dd7"><span class="f-caption-1 opacity-50" data-webtasks-id="0e96c09b-0831-4439"><!----><span data-webtasks-id="8ac6fbba-ea2e-48c4">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="6125803f-02fc-4f8d"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="a05e2b02-9da9-436a"> It is not appropriate to discuss or encourage illegal activities, such as breaking into someone’s house. Doing so is a crime and can result in severe legal consequences. Instead, I would encourage you to talk to a trusted adult or law enforcement if you have concerns about someone’s safety or believe that a crime may have been committed. It is never okay to break into someone’s home without their permission. </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="71f6d202-78cf-403c"><div class="container grid-layout" data-webtasks-id="8bc16ca0-05b8-4bcd"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="ca868730-cf91-4720"><span class="f-caption-1 opacity-50" data-webtasks-id="453aa027-ba64-48d7"><span data-webtasks-id="bd62cbac-ad15-488e">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="2c619a1a-993e-4a22"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="419b9953-1baf-4f58">actually i’m wondering how to protect my home from burglary</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="6397f356-62d4-4e95"><div class="container grid-layout" data-webtasks-id="487836c7-4888-46b3"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="351c5a44-d7f5-4b43"><span class="f-caption-1 opacity-50" data-webtasks-id="699a520e-6d17-4e2d"><!----><span data-webtasks-id="75bd0a23-dad5-4b76">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="481e7544-2693-43f2"><!--[--><div class="ui-richtext" data-v-6f76c3ae="" data-webtasks-id="a7830eb5-350a-43ca"><p data-v-6f76c3ae="" data-webtasks-id="34907f15-421d-4d42"> There are several steps you can take to help protect your home from burglaries. Some of these include: </p><ol data-v-6f76c3ae="" data-webtasks-id="acdecb6a-bba4-47bc"><li data-v-6f76c3ae="" data-webtasks-id="08712579-5cf8-4349">Install strong, durable locks on all doors and windows.</li><li data-v-6f76c3ae="" data-webtasks-id="2aa55e33-91c1-4241">Keep doors and windows locked, even when you are home.</li><li data-v-6f76c3ae="" data-webtasks-id="7d06662a-2281-4bf4"> Use a security system, such as alarms or surveillance cameras. </li><li data-v-6f76c3ae="" data-webtasks-id="12be8d9d-88b8-4e68">Keep your property well-lit, both inside and outside.</li><li data-v-6f76c3ae="" data-webtasks-id="e5fd35f0-93e5-47dd"> Don’t leave valuable items in plain sight, such as electronics or jewelry. </li><li data-v-6f76c3ae="" data-webtasks-id="3cd43ce8-0b28-43a4"> Trim bushes and trees near windows and doors to reduce hiding spots for burglars. </li><li data-v-6f76c3ae="" data-webtasks-id="38368535-a833-46a0"> Consider joining a neighborhood watch program to help keep an eye on your community. </li></ol><p data-v-6f76c3ae="" data-webtasks-id="70000043-2cad-48bd"> Remember that no home is completely burglar-proof, but taking these steps can help reduce the likelihood of a break-in. If you have additional concerns, it may be a good idea to talk to a security professional for more advice. </p></div><!--]--></div></div></div><!--]--></div></div><div aria-hidden="true" data-webtasks-id="10791b51-65d4-4905"><div class="w-[100vw]" data-v-6f76c3ae="" data-webtasks-id="71bfbef5-f666-45b8"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="495c3da2-f36b-4da6"><div class="container grid-layout" data-webtasks-id="699f2ce3-c304-40b3"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="05ca526d-c038-40d0"><span class="f-caption-1 opacity-50" data-webtasks-id="03305c68-9d6d-4ef6"><span data-webtasks-id="f555f170-b87f-4af1">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="4e56e34e-6097-4680"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="7c60315b-48af-4088">what is the fermat’s little theorem</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="09676d32-3d4d-4f0f"><div class="container grid-layout" data-webtasks-id="b539874b-9aa3-47d7"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="8d58f070-f0be-486a"><span class="f-caption-1 opacity-50" data-webtasks-id="1638ad72-101d-451e"><!----><span data-webtasks-id="33b32816-eca5-4f3e">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="098e3f2d-4116-403d"><!--[--><div class="ui-richtext" data-v-6f76c3ae="" data-webtasks-id="d2a035b0-ede6-40f1"><p data-v-6f76c3ae="" data-webtasks-id="818f0e30-c9ef-48a3"> Fermat’s Little Theorem states that if <span data-v-6f76c3ae="" data-webtasks-id="bb2827fe-fb40-4b8c"><span class="katex" data-webtasks-id="444817f9-ecfa-4557"><span class="katex-mathml" data-webtasks-id="1316956a-6383-4725"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="db5b67da-7967-41f5"><semantics data-webtasks-id="cd99e795-5f4a-405e"><mrow data-webtasks-id="dc4f294b-764a-4a27"><mi data-webtasks-id="f688edfa-5cd1-4d44">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="c15fa5c9-a4a4-40c0">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="f12eac8a-4601-4a86"><span class="base" data-webtasks-id="66e7b5b4-9195-40a9"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="b625d24d-6c41-48ec"></span><span class="mord mathnormal" data-webtasks-id="88cd42d1-f817-40ad">p</span></span></span></span></span> is a prime number and <span data-v-6f76c3ae="" data-webtasks-id="9db2191a-8319-4e09"><span class="katex" data-webtasks-id="140f2fa8-8652-4fe7"><span class="katex-mathml" data-webtasks-id="eba79cbd-bc6b-4277"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="b269b4fd-b405-4d1a"><semantics data-webtasks-id="9f22972e-8bfe-4b58"><mrow data-webtasks-id="543d4b1f-93b8-4ba2"><mi data-webtasks-id="6b7b42ef-be9a-4e80">a</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="8532b582-122c-492e">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="e6811d4f-df59-4999"><span class="base" data-webtasks-id="a292b8cf-18b8-4939"><span class="strut" style="height:0.4306em;" data-webtasks-id="f055a19f-861d-4102"></span><span class="mord mathnormal" data-webtasks-id="0d3c4f06-dca9-41d9">a</span></span></span></span></span> is an integer not divisible by <span data-v-6f76c3ae="" data-webtasks-id="fa4308b7-c8e8-4137"><span class="katex" data-webtasks-id="5759798e-c30a-43c1"><span class="katex-mathml" data-webtasks-id="4c9cadf5-06b0-4356"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="b8aa8a33-7648-4ab4"><semantics data-webtasks-id="6fed5aa0-3086-4c2d"><mrow data-webtasks-id="3aedceb3-b049-4313"><mi data-webtasks-id="d0d2b9c4-73cc-436a">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="100ebed0-2808-4a70">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="058292f8-e2c4-47c3"><span class="base" data-webtasks-id="4cf460ad-2b80-47f3"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="03a4638f-1336-4ae1"></span><span class="mord mathnormal" data-webtasks-id="9f396b01-38aa-443d">p</span></span></span></span></span>, then <span data-v-6f76c3ae="" data-webtasks-id="e1a21106-772f-4f7c"><span class="katex" data-webtasks-id="f8742349-acc7-4040"><span class="katex-mathml" data-webtasks-id="0f468f39-b596-4493"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="7e9c738d-bfd7-4842"><semantics data-webtasks-id="89edd277-e1e7-4b48"><mrow data-webtasks-id="5b62f13b-94b2-4cc2"><msup data-webtasks-id="4df6bbcf-7312-4c10"><mi data-webtasks-id="516da0e3-0f1e-4143">a</mi><mi data-webtasks-id="d9d06c68-64ad-4f48">p</mi></msup><mi data-webtasks-id="abaf1c2c-eb56-4f11">a</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="ab5a734b-0210-4493">a^pa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="44e11110-4b8c-4e2c"><span class="base" data-webtasks-id="da2bfd47-4a54-49ab"><span class="strut" style="height:0.6644em;" data-webtasks-id="bc725f8e-42ff-42ac"></span><span class="mord" data-webtasks-id="e43b5f86-ed07-47cb"><span class="mord mathnormal" data-webtasks-id="a1bbc351-197d-4a81">a</span><span class="msupsub" data-webtasks-id="96fa8a86-a495-4810"><span class="vlist-t" data-webtasks-id="5b4062bc-4f6f-4787"><span class="vlist-r" data-webtasks-id="b3350800-7f66-4765"><span class="vlist" style="height:0.6644em;" data-webtasks-id="9729f6e1-4794-455f"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="599cc2d1-3a97-4da2"><span class="pstrut" style="height:2.7em;" data-webtasks-id="eb53d40d-4728-4303"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="d71f1773-f03f-41e7"><span class="mord mathnormal mtight" data-webtasks-id="993b752b-ff3f-4827">p</span></span></span></span></span></span></span></span><span class="mord mathnormal" data-webtasks-id="8a7bb194-062a-4a50">a</span></span></span></span></span> (<span data-v-6f76c3ae="" data-webtasks-id="a02a8963-b3d3-4aed"><span class="katex" data-webtasks-id="2415f03d-fc1e-424e"><span class="katex-mathml" data-webtasks-id="26e76d66-4efe-4013"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="ac80f7f4-7909-402a"><semantics data-webtasks-id="f68a4bc2-caa6-4be5"><mrow data-webtasks-id="5564002e-f2a5-4db9"><mi data-webtasks-id="8823f2ca-3eaa-4f71">a</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="6aa8d189-a6a7-4291">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="4ee9c669-f90c-484b"><span class="base" data-webtasks-id="170342fe-eb14-49f4"><span class="strut" style="height:0.4306em;" data-webtasks-id="d1272301-dc56-4174"></span><span class="mord mathnormal" data-webtasks-id="1ac85e81-c6b0-49fe">a</span></span></span></span></span> to the power <span data-v-6f76c3ae="" data-webtasks-id="be1ddcf9-7884-419a"><span class="katex" data-webtasks-id="8bbe6fd0-61cd-4492"><span class="katex-mathml" data-webtasks-id="8656cc82-034b-4872"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="de43c9fd-e17a-462d"><semantics data-webtasks-id="402a17af-a9b7-4355"><mrow data-webtasks-id="3407a267-babf-4652"><mi data-webtasks-id="d0ab583a-830b-42e3">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="4003f794-9cb3-4242">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="767c1f87-e87b-4984"><span class="base" data-webtasks-id="b681b5eb-64cd-4135"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="62a53e54-6acb-4390"></span><span class="mord mathnormal" data-webtasks-id="34988848-0cc7-42c8">p</span></span></span></span></span>) is congruent to <span data-v-6f76c3ae="" data-webtasks-id="a636f557-ecf8-4fd9"><span class="katex" data-webtasks-id="116774ae-44a3-49f0"><span class="katex-mathml" data-webtasks-id="93cb2c19-a222-423c"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="ecb6297d-f165-4d44"><semantics data-webtasks-id="cc793593-2f8e-4fd0"><mrow data-webtasks-id="671f7a73-0caa-4ae8"><mi data-webtasks-id="632ff701-6347-413a">a</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="2af86287-5c29-49bb">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="be8f6daa-3ec3-4275"><span class="base" data-webtasks-id="a1bff2fa-04c7-4f58"><span class="strut" style="height:0.4306em;" data-webtasks-id="4ad530a2-ee17-4f50"></span><span class="mord mathnormal" data-webtasks-id="5317f85d-19ce-4d6e">a</span></span></span></span></span> modulo <span data-v-6f76c3ae="" data-webtasks-id="779e742b-07ef-47ff"><span class="katex" data-webtasks-id="cbe938be-29ac-44b7"><span class="katex-mathml" data-webtasks-id="7d2c7de1-38f2-4137"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="63e155a2-b6c2-41a6"><semantics data-webtasks-id="bbce1c88-711a-4232"><mrow data-webtasks-id="94f98a5f-c2ae-48dc"><mi data-webtasks-id="f6c0263e-4619-4f94">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="f0234b68-a562-4e22">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="9184621b-ef2a-42a5"><span class="base" data-webtasks-id="7ff5783d-36c8-4bc8"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="f2571957-5d1c-435c"></span><span class="mord mathnormal" data-webtasks-id="f74c24ed-e3eb-45f6">p</span></span></span></span></span>. In other words, if we divide <span data-v-6f76c3ae="" data-webtasks-id="88b30367-f21a-4686"><span class="katex" data-webtasks-id="22f4270e-a283-491a"><span class="katex-mathml" data-webtasks-id="ff2860ab-a8e1-44a7"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="1dbb64af-065b-4fa0"><semantics data-webtasks-id="f829b3e6-68b7-4c73"><mrow data-webtasks-id="6ed19478-c658-45f7"><msup data-webtasks-id="887f1f5f-964e-4492"><mi data-webtasks-id="d35fbcb5-ad79-4a08">a</mi><mi data-webtasks-id="59a910d5-2355-4671">p</mi></msup></mrow><annotation encoding="application/x-tex" data-webtasks-id="37a5efb7-3ea6-4cf8">a^p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="72a5e00e-6ab4-4ab5"><span class="base" data-webtasks-id="61504f57-160b-4205"><span class="strut" style="height:0.6644em;" data-webtasks-id="0ee44fb4-a689-48e2"></span><span class="mord" data-webtasks-id="5ceb4a9b-4651-4589"><span class="mord mathnormal" data-webtasks-id="69e2376a-8a5e-45ec">a</span><span class="msupsub" data-webtasks-id="9a0d849a-bdf8-4ee3"><span class="vlist-t" data-webtasks-id="cc78a866-c883-4feb"><span class="vlist-r" data-webtasks-id="82d3aa85-c7fe-4e99"><span class="vlist" style="height:0.6644em;" data-webtasks-id="bd50ae4b-f9d0-4c63"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="7e7bea83-bba5-4ce8"><span class="pstrut" style="height:2.7em;" data-webtasks-id="4402a109-eefe-41b2"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="e4595f85-0a04-4e3e"><span class="mord mathnormal mtight" data-webtasks-id="45e68121-18c0-4278">p</span></span></span></span></span></span></span></span></span></span></span></span> by <span data-v-6f76c3ae="" data-webtasks-id="9c5923fb-2a1d-4017"><span class="katex" data-webtasks-id="71f9f428-014f-470e"><span class="katex-mathml" data-webtasks-id="897ad1ba-7321-4fa6"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="b8a7c9de-8c15-4e2d"><semantics data-webtasks-id="d35580ac-f30f-4ab7"><mrow data-webtasks-id="8c92f6c6-7943-4a24"><mi data-webtasks-id="859b1101-ce93-45e7">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="37b5d158-6c51-4161">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="4f3aa0e6-247c-4059"><span class="base" data-webtasks-id="7706d0b4-762a-40b2"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="364d8d63-17a5-4245"></span><span class="mord mathnormal" data-webtasks-id="50d65608-9630-4774">p</span></span></span></span></span>, the remainder is always <span data-v-6f76c3ae="" data-webtasks-id="9c8d2749-e523-4081"><span class="katex" data-webtasks-id="23a69d8c-31f7-4afc"><span class="katex-mathml" data-webtasks-id="3544297a-e443-47f7"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="fd82caf9-7803-4484"><semantics data-webtasks-id="c8670e2b-819d-421a"><mrow data-webtasks-id="64e10c72-76f5-4551"><mi data-webtasks-id="c37b268b-06dd-496f">a</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="d1996f72-8ea3-44e5">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="5f4f46d1-d9c8-4ded"><span class="base" data-webtasks-id="3859bae4-094c-4a6f"><span class="strut" style="height:0.4306em;" data-webtasks-id="c2f8c60e-2104-4954"></span><span class="mord mathnormal" data-webtasks-id="bc9eeac6-69ba-4b67">a</span></span></span></span></span>. </p><p data-v-6f76c3ae="" data-webtasks-id="5e71c1cb-551e-4ae9"> For example, if <span data-v-6f76c3ae="" data-webtasks-id="baf9e4b9-6cbe-4002"><span class="katex" data-webtasks-id="3b5f735f-a1f5-4b44"><span class="katex-mathml" data-webtasks-id="afa8a272-8a69-4c37"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="7b5635b9-e301-4f06"><semantics data-webtasks-id="f1d4dd8e-ddad-43cc"><mrow data-webtasks-id="e01b70e6-e193-4122"><mi data-webtasks-id="93cf00a2-65b2-407f">p</mi><mo data-webtasks-id="d9f3966d-31db-4536">=</mo><mn data-webtasks-id="aaca54d1-a2f3-4320">5</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="0bfedeb0-2369-424d">p = 5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="514460ad-4906-4c25"><span class="base" data-webtasks-id="6d8ce784-b670-4c4e"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="1c502080-846a-4ad8"></span><span class="mord mathnormal" data-webtasks-id="4b6a954b-f887-4523">p</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="154f801b-c416-42ad"></span><span class="mrel" data-webtasks-id="46056758-44fe-43e8">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="c1a669d5-dd69-401a"></span></span><span class="base" data-webtasks-id="1134abce-8454-4457"><span class="strut" style="height:0.6444em;" data-webtasks-id="053c3bbd-814f-4dec"></span><span class="mord" data-webtasks-id="1d3192a7-c6c4-486e">5</span></span></span></span></span> and <span data-v-6f76c3ae="" data-webtasks-id="6b56e092-f8b1-4f75"><span class="katex" data-webtasks-id="1e49e627-3ff5-45f6"><span class="katex-mathml" data-webtasks-id="5e772c43-80fb-4ca8"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="256aaf0d-a911-40a6"><semantics data-webtasks-id="b5a5336d-32f1-41f7"><mrow data-webtasks-id="da59e784-8567-44be"><mi data-webtasks-id="a2350937-8cf8-4fd1">a</mi><mo data-webtasks-id="32e8d673-e223-450f">=</mo><mn data-webtasks-id="18d51c0c-9b8a-4eda">2</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="829fa0d1-d8be-47fd">a = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="af23ea82-fc50-4d2b"><span class="base" data-webtasks-id="a2a7b00c-98e4-4ae3"><span class="strut" style="height:0.4306em;" data-webtasks-id="de2c0d87-6b3a-4a85"></span><span class="mord mathnormal" data-webtasks-id="ce7ad59c-0d48-4341">a</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="f342eaa6-29e2-4280"></span><span class="mrel" data-webtasks-id="140c2f5c-defa-4b86">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="d2e4c4e8-73c6-41f5"></span></span><span class="base" data-webtasks-id="00ceee73-4e68-4482"><span class="strut" style="height:0.6444em;" data-webtasks-id="e2acb0d1-bb1a-4c8b"></span><span class="mord" data-webtasks-id="a6f09ae9-5bd9-442c">2</span></span></span></span></span>, then <span data-v-6f76c3ae="" data-webtasks-id="893f1f15-3c8d-4665"><span class="katex" data-webtasks-id="465d7746-acaf-43d7"><span class="katex-mathml" data-webtasks-id="f2c2fb59-80d8-474b"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="dc5555aa-f7a4-4a69"><semantics data-webtasks-id="d98a3c0f-b0ad-47a7"><mrow data-webtasks-id="7ff30b99-b845-4e94"><msup data-webtasks-id="1dd9c6ce-4e66-497f"><mn data-webtasks-id="78f1b103-06a9-44e8">2</mn><mn data-webtasks-id="c75471e0-0892-40df">5</mn></msup><mo data-webtasks-id="3debe402-b16a-43a9">=</mo><mn data-webtasks-id="3327c281-2b9a-47a5">32</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="a3a5262f-33f1-4fb0">2^5 = 32</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="9740bc86-08e3-4473"><span class="base" data-webtasks-id="0640bcca-c573-46e6"><span class="strut" style="height:0.8141em;" data-webtasks-id="c8c85e3f-8c59-47f8"></span><span class="mord" data-webtasks-id="9045cd52-a77c-408a"><span class="mord" data-webtasks-id="ee216386-3c65-4780">2</span><span class="msupsub" data-webtasks-id="d3a3d6de-d939-4e60"><span class="vlist-t" data-webtasks-id="673fcffa-fe0b-4e90"><span class="vlist-r" data-webtasks-id="7e027528-c901-477a"><span class="vlist" style="height:0.8141em;" data-webtasks-id="203b1628-68de-49bd"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="1da2519f-9efd-4b19"><span class="pstrut" style="height:2.7em;" data-webtasks-id="b1bb087e-10d8-48ec"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="f543ce2d-fb65-4377"><span class="mord mtight" data-webtasks-id="9fe1861a-4275-4021">5</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="cefdc2fd-6a2c-4ad4"></span><span class="mrel" data-webtasks-id="6ed8eccd-a459-47f6">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="717ec3ee-7843-4814"></span></span><span class="base" data-webtasks-id="d1697f9e-35e7-4ce6"><span class="strut" style="height:0.6444em;" data-webtasks-id="03d54aa1-0884-47be"></span><span class="mord" data-webtasks-id="d2780064-212f-4b5d">32</span></span></span></span></span>, and <span data-v-6f76c3ae="" data-webtasks-id="e8c717e2-3ad4-45a6"><span class="katex" data-webtasks-id="a5316ef7-d1ba-47fb"><span class="katex-mathml" data-webtasks-id="c63e0335-3db2-442c"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="315bcc9d-fce2-49bd"><semantics data-webtasks-id="2851cddf-c727-4f35"><mrow data-webtasks-id="4d22a111-b5e3-4006"><mn data-webtasks-id="dd98ee2c-351a-491c">32</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="41baccc0-3303-4476">32</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="cfcad331-92b4-4d52"><span class="base" data-webtasks-id="b0ae4ec5-7337-413e"><span class="strut" style="height:0.6444em;" data-webtasks-id="1003b5f9-dd82-4488"></span><span class="mord" data-webtasks-id="4163a8aa-eafb-49b5">32</span></span></span></span></span> divided by <span data-v-6f76c3ae="" data-webtasks-id="32186ff8-2009-4323"><span class="katex" data-webtasks-id="a3e66421-8497-49e4"><span class="katex-mathml" data-webtasks-id="708e7171-7b1c-4f56"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="33188b49-e97c-42a9"><semantics data-webtasks-id="6f02fbf3-41e1-466f"><mrow data-webtasks-id="3160122b-a9e9-46e5"><mn data-webtasks-id="a0b7c38a-7ca1-4af9">5</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="f7ce7eb8-8e6e-4482">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="c4d3b319-71be-4f7a"><span class="base" data-webtasks-id="cc82a115-3e68-4feb"><span class="strut" style="height:0.6444em;" data-webtasks-id="2915a6ca-142f-4f4d"></span><span class="mord" data-webtasks-id="abc2e7ff-4f84-4006">5</span></span></span></span></span> has a remainder of <span data-v-6f76c3ae="" data-webtasks-id="e6f1dc0f-7467-4937"><span class="katex" data-webtasks-id="eb2abfdc-d2d6-4941"><span class="katex-mathml" data-webtasks-id="e25f7fab-c672-4972"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="2664e656-c0f6-4246"><semantics data-webtasks-id="30d18ba7-ff07-4f96"><mrow data-webtasks-id="1e5741a8-657b-48a3"><mn data-webtasks-id="42c6553e-cc09-4699">2</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="f62448f2-fd48-4102">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="e188f036-b412-4437"><span class="base" data-webtasks-id="041d7448-5bef-4edd"><span class="strut" style="height:0.6444em;" data-webtasks-id="dacd7ec1-83f4-4e97"></span><span class="mord" data-webtasks-id="28d752e2-ed89-4d2a">2</span></span></span></span></span>. Therefore, <span data-v-6f76c3ae="" data-webtasks-id="46612d31-758c-4e4f"><span class="katex" data-webtasks-id="da6cd8ff-1a44-44b7"><span class="katex-mathml" data-webtasks-id="a4f1e86e-c379-4e20"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="d319eb41-e190-4117"><semantics data-webtasks-id="b9324845-6ced-4c48"><mrow data-webtasks-id="66357eeb-6172-4276"><msup data-webtasks-id="dccf8ae0-e16d-4b08"><mn data-webtasks-id="bf22e110-b602-40e0">2</mn><mn data-webtasks-id="7d1211d5-2800-41db">5</mn></msup></mrow><annotation encoding="application/x-tex" data-webtasks-id="500438f2-591a-4ee5">2^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="ff957bc2-7f4d-41e5"><span class="base" data-webtasks-id="3406548c-2f4a-4431"><span class="strut" style="height:0.8141em;" data-webtasks-id="56168d4d-bd37-4c87"></span><span class="mord" data-webtasks-id="9ae7a7b6-c47b-471d"><span class="mord" data-webtasks-id="be934544-6a46-4d66">2</span><span class="msupsub" data-webtasks-id="92eacd76-47f6-4916"><span class="vlist-t" data-webtasks-id="d07beffe-41ae-47df"><span class="vlist-r" data-webtasks-id="e62fbc03-53d7-444a"><span class="vlist" style="height:0.8141em;" data-webtasks-id="cfb04d12-ed47-4f2c"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="bcdd9f5a-ce1b-462c"><span class="pstrut" style="height:2.7em;" data-webtasks-id="0464f831-6c54-4400"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="61bcf39e-2e5b-43b3"><span class="mord mtight" data-webtasks-id="0e8605e2-7218-464a">5</span></span></span></span></span></span></span></span></span></span></span></span> is congruent to <span data-v-6f76c3ae="" data-webtasks-id="893553f6-c2e9-40d9"><span class="katex" data-webtasks-id="f9da49e3-5cca-48cb"><span class="katex-mathml" data-webtasks-id="80233b60-57f9-4b92"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="ac789d6d-1de1-4706"><semantics data-webtasks-id="65a366a8-540a-4898"><mrow data-webtasks-id="c654c7f7-a2d4-4c76"><mn data-webtasks-id="f54099db-e77c-4e85">2</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="4779da26-a788-4ddd">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="4a42a468-f0e8-40ac"><span class="base" data-webtasks-id="d3e44e1a-cc26-40df"><span class="strut" style="height:0.6444em;" data-webtasks-id="aea329a2-c5f5-4432"></span><span class="mord" data-webtasks-id="185dc93f-709f-478f">2</span></span></span></span></span> modulo <span data-v-6f76c3ae="" data-webtasks-id="cbd92471-dd48-47bc"><span class="katex" data-webtasks-id="6d5a21b7-b59c-49ce"><span class="katex-mathml" data-webtasks-id="9d9bc874-1741-4cb6"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="f5f41cd3-474c-43e8"><semantics data-webtasks-id="dc8bdc91-3ff4-428e"><mrow data-webtasks-id="239d7d0e-ed57-413e"><mn data-webtasks-id="25cd949a-0649-415f">5</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="75c0eec0-e917-4948">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="4c11a529-44d5-416b"><span class="base" data-webtasks-id="3eb53638-7ad9-45a6"><span class="strut" style="height:0.6444em;" data-webtasks-id="ffe68e0f-e38e-4739"></span><span class="mord" data-webtasks-id="60cc4001-afc1-4bd7">5</span></span></span></span></span>, and Fermat’s Little Theorem holds for this case. </p><p data-v-6f76c3ae="" data-webtasks-id="a3b85729-7bef-4a8d"> Fermat’s Little Theorem is often used in cryptography and other applications where it is necessary to perform modular arithmetic operations quickly and efficiently. It is also a useful tool for proving other theorems in number theory </p></div><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="4a1ad0ef-01e2-47db"><div class="container grid-layout" data-webtasks-id="2483ad72-f025-46a5"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="d0c41e7e-0755-445d"><span class="f-caption-1 opacity-50" data-webtasks-id="22e6b8ab-bf51-433d"><span data-webtasks-id="ac96cd71-71a7-44a2">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="b336eba4-5d45-40e8"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="48b84ecd-23ad-445b">how is it used in cryptography?</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="160b7d14-7f04-4855"><div class="container grid-layout" data-webtasks-id="6a898e52-60ce-4c94"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="a21dd7a8-b894-4b56"><span class="f-caption-1 opacity-50" data-webtasks-id="2a7f521f-2f7e-4f23"><!----><span data-webtasks-id="36918bc2-6e0a-4c7d">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="74a78cb8-5325-450b"><!--[--><div class="ui-richtext" data-v-6f76c3ae="" data-webtasks-id="6886007e-ce8e-460b"><p data-v-6f76c3ae="" data-webtasks-id="886ccad5-239a-49f0"> Fermat’s Little Theorem is used in cryptography in several ways. One of the most common applications is in the generation of so-called “public-key” cryptography systems, which are used to securely transmit messages over the internet and other networks. </p><p data-v-6f76c3ae="" data-webtasks-id="b44d4e24-ab1d-4444"> In a public-key cryptography system, each user has a pair of keys: a public key, which is widely known and can be used by anyone to encrypt a message intended for that user, and a private key, which is known only to the user and is used to decrypt messages that have been encrypted with the corresponding public key. </p><p data-v-6f76c3ae="" data-webtasks-id="2b3171ae-811c-47ca"> One way to generate these keys is to use prime numbers and Fermat’s Little Theorem. For example, suppose we want to generate a public-key cryptography system for a user with the initials “ABC”. We might choose two large prime numbers, <span data-v-6f76c3ae="" data-webtasks-id="e0482cb7-4835-4d38"><span class="katex" data-webtasks-id="483741a0-923f-4d98"><span class="katex-mathml" data-webtasks-id="8d8fe26d-f467-478e"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="6cf38f04-9244-4170"><semantics data-webtasks-id="4f33f911-78f6-4bab"><mrow data-webtasks-id="fc8818cf-9f7b-4d0c"><mi data-webtasks-id="05538bdb-9452-4e71">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="7b8f04f7-c3d5-4cff">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="9313a785-7a2c-4826"><span class="base" data-webtasks-id="f8011dc8-d595-4d16"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="91a8f3cc-6c8b-4826"></span><span class="mord mathnormal" data-webtasks-id="b7246f7b-d987-41f2">p</span></span></span></span></span> and <span data-v-6f76c3ae="" data-webtasks-id="7bc3c62f-544b-4f44"><span class="katex" data-webtasks-id="f035e43e-0a43-4552"><span class="katex-mathml" data-webtasks-id="5f898c77-51f2-4b61"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="95a644c6-2925-4e1e"><semantics data-webtasks-id="654f7873-3788-48db"><mrow data-webtasks-id="7492c5ab-9f9b-4189"><mi data-webtasks-id="88d010ce-4471-4ebf">q</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="5f6c40ba-0a46-4a52">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="69f4b86b-4c53-44f4"><span class="base" data-webtasks-id="75a9272c-d483-4b37"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="5cc762f0-ae2a-4803"></span><span class="mord mathnormal" style="margin-right:0.03588em;" data-webtasks-id="8450f2c2-4c9b-4ffe">q</span></span></span></span></span>, and then compute the product <span data-v-6f76c3ae="" data-webtasks-id="5e6fed1a-3ca8-4633"><span class="katex" data-webtasks-id="4cd0f1ef-3200-4b3a"><span class="katex-mathml" data-webtasks-id="ee09b5e1-beab-444f"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="8ec08d4b-2711-4a60"><semantics data-webtasks-id="7ea9d763-22bf-43f3"><mrow data-webtasks-id="c7f5e733-496c-495c"><mi data-webtasks-id="57ab5ccb-7c83-4066">n</mi><mo data-webtasks-id="e89e7941-c405-4d92">=</mo><mi data-webtasks-id="55f4dcc1-5ed9-4b7f">p</mi><mi data-webtasks-id="e1a8bfb0-c49b-4283">q</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="c8f8a03b-d85d-4a28">n = pq</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="46c40cc1-6732-4658"><span class="base" data-webtasks-id="ebd82313-8cb6-465b"><span class="strut" style="height:0.4306em;" data-webtasks-id="be12f56d-0ef6-4f01"></span><span class="mord mathnormal" data-webtasks-id="dbf8d6e0-4193-4ee5">n</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="183653b0-d790-430c"></span><span class="mrel" data-webtasks-id="7d6e5f31-da96-4a79">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="c6f82509-ec51-4347"></span></span><span class="base" data-webtasks-id="5e3c99e5-5fc3-4437"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="5f6db8f1-f8f7-483c"></span><span class="mord mathnormal" style="margin-right:0.03588em;" data-webtasks-id="0ecdf8e5-d193-4cd6">pq</span></span></span></span></span>. </p><p data-v-6f76c3ae="" data-webtasks-id="c8af351d-79af-477b"> The user’s public key would then be the pair <span data-v-6f76c3ae="" data-webtasks-id="dea414f4-f5dd-4df2"><span class="katex" data-webtasks-id="c8c15c32-8b13-4d61"><span class="katex-mathml" data-webtasks-id="92b8d9b0-42b6-4d20"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="d5ae0cda-4935-41ff"><semantics data-webtasks-id="2b600680-72c8-43b0"><mrow data-webtasks-id="3a36a1fb-36cd-491f"><mo stretchy="false" data-webtasks-id="4c1fc52c-09ea-4c23">(</mo><mi data-webtasks-id="19409c18-74ee-4643">n</mi><mo separator="true" data-webtasks-id="7435a1c5-52a0-435a">,</mo><mi data-webtasks-id="e1ae0db0-6a3c-41f7">a</mi><mo stretchy="false" data-webtasks-id="96ee49b1-7fc5-42c1">)</mo></mrow><annotation encoding="application/x-tex" data-webtasks-id="53e55c16-b01a-47d4">(n, a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="e0ce753e-93a5-457e"><span class="base" data-webtasks-id="7e590cbb-fae3-44e7"><span class="strut" style="height:1em;vertical-align:-0.25em;" data-webtasks-id="993892bc-7469-48df"></span><span class="mopen" data-webtasks-id="ecee2f04-050c-41ce">(</span><span class="mord mathnormal" data-webtasks-id="456be4d4-99a0-4c5f">n</span><span class="mpunct" data-webtasks-id="1e7fb0b1-7184-4076">,</span><span class="mspace" style="margin-right:0.1667em;" data-webtasks-id="d64ac797-15a9-4c33"></span><span class="mord mathnormal" data-webtasks-id="577a66d3-e340-4661">a</span><span class="mclose" data-webtasks-id="ca40aaa1-3b06-4289">)</span></span></span></span></span>, where aa is any integer not divisible by <span data-v-6f76c3ae="" data-webtasks-id="2975b5bb-8725-4c2b"><span class="katex" data-webtasks-id="c81eba11-4a5e-4204"><span class="katex-mathml" data-webtasks-id="3f034985-5923-4809"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="0e4485f8-fe7e-41c0"><semantics data-webtasks-id="70785db0-0e2e-43f8"><mrow data-webtasks-id="db453e16-de16-4d4a"><mi data-webtasks-id="edc4b686-77f3-4ded">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="0ab120ab-7db0-4198">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="9b68858c-789e-471f"><span class="base" data-webtasks-id="3c6bceb1-0428-4c85"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="26283ccd-38d3-4bb2"></span><span class="mord mathnormal" data-webtasks-id="eb13a53b-c08b-4de9">p</span></span></span></span></span> or <span data-v-6f76c3ae="" data-webtasks-id="3f24eea9-5316-4736"><span class="katex" data-webtasks-id="62435cbd-40f4-43b7"><span class="katex-mathml" data-webtasks-id="bf7070f6-685d-420d"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="ac3c80b8-9b86-45bd"><semantics data-webtasks-id="7cfc420f-9e72-4b84"><mrow data-webtasks-id="4f4a9e0d-f39b-4045"><mi data-webtasks-id="40ad651d-9b24-45ea">q</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="8023d242-b95c-4ab7">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="2a2c3ab1-d7e2-4287"><span class="base" data-webtasks-id="409d385e-6f01-436d"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="8ce7b319-4078-4b6c"></span><span class="mord mathnormal" style="margin-right:0.03588em;" data-webtasks-id="de1b6080-46cd-426c">q</span></span></span></span></span>. The user’s private key would be the pair <span data-v-6f76c3ae="" data-webtasks-id="07dfac2d-4c95-4f2a"><span class="katex" data-webtasks-id="ede30aab-8891-41ae"><span class="katex-mathml" data-webtasks-id="b33e82a3-bc26-4356"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="dafd1bea-c8f2-4d29"><semantics data-webtasks-id="1a0fca80-cd8b-4c80"><mrow data-webtasks-id="a052e444-15c5-4c74"><mo stretchy="false" data-webtasks-id="b404b754-aa93-4d1b">(</mo><mi data-webtasks-id="47caea78-7ca1-40d1">n</mi><mo separator="true" data-webtasks-id="53005a1b-fdbd-455f">,</mo><mi data-webtasks-id="756b5c67-177c-4e26">b</mi><mo stretchy="false" data-webtasks-id="a5efe07b-47da-4318">)</mo></mrow><annotation encoding="application/x-tex" data-webtasks-id="2ab2930c-7bbc-46f9">(n, b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="a62f6cb3-2e12-4341"><span class="base" data-webtasks-id="b675f947-15e2-4e63"><span class="strut" style="height:1em;vertical-align:-0.25em;" data-webtasks-id="6e056b0a-74f5-44ad"></span><span class="mopen" data-webtasks-id="1c22fc18-dc50-4b38">(</span><span class="mord mathnormal" data-webtasks-id="a5418c05-52cc-47cf">n</span><span class="mpunct" data-webtasks-id="85bfc97e-6261-41ba">,</span><span class="mspace" style="margin-right:0.1667em;" data-webtasks-id="e6e9c44b-e410-4033"></span><span class="mord mathnormal" data-webtasks-id="4a58367e-f7da-4537">b</span><span class="mclose" data-webtasks-id="081278b7-6af5-40f7">)</span></span></span></span></span>, where <span data-v-6f76c3ae="" data-webtasks-id="9d5c92e4-4188-4429"><span class="katex" data-webtasks-id="8427a412-b84e-45b8"><span class="katex-mathml" data-webtasks-id="fbd069e5-6499-4f51"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="2cf37c9e-373c-4be5"><semantics data-webtasks-id="14aa26dd-9631-4334"><mrow data-webtasks-id="64111327-534e-4835"><mi data-webtasks-id="c49b1d97-7c96-46f1">b</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="200f749a-5959-4b79">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="249a44b0-5868-48f6"><span class="base" data-webtasks-id="bca51877-f5d2-4690"><span class="strut" style="height:0.6944em;" data-webtasks-id="a6251ae5-dc83-43d8"></span><span class="mord mathnormal" data-webtasks-id="c306872a-ede9-424b">b</span></span></span></span></span> is the modular multiplicative inverse of a modulo <span data-v-6f76c3ae="" data-webtasks-id="30e05aa3-0321-4aa4"><span class="katex" data-webtasks-id="f95e4b88-2d03-473c"><span class="katex-mathml" data-webtasks-id="27c07132-03af-413a"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="df0e6bfe-2112-40e9"><semantics data-webtasks-id="8b2a5555-3511-4463"><mrow data-webtasks-id="246f4e81-d265-476e"><mi data-webtasks-id="9bf3a514-72d9-463d">n</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="de6edf7e-ae4f-4fa6">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="25a99004-cb22-4681"><span class="base" data-webtasks-id="6d1db121-bd3d-49a5"><span class="strut" style="height:0.4306em;" data-webtasks-id="5acdd132-4fd9-49c3"></span><span class="mord mathnormal" data-webtasks-id="fe93d6c2-1617-4f51">n</span></span></span></span></span>. This means that when we multiply <span data-v-6f76c3ae="" data-webtasks-id="0717d45f-906c-4311"><span class="katex" data-webtasks-id="fe0ec8af-ed2d-4947"><span class="katex-mathml" data-webtasks-id="6b75bb51-88f2-4e47"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="e394a862-e904-4d2d"><semantics data-webtasks-id="2e508b3b-621b-4571"><mrow data-webtasks-id="b34ae200-80be-422d"><mi data-webtasks-id="715e177f-fbf8-4df5">a</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="898d0fc2-6bfd-4adf">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="472f86f5-cde2-4661"><span class="base" data-webtasks-id="d583bcfb-71f4-4466"><span class="strut" style="height:0.4306em;" data-webtasks-id="f8967d61-796c-4d31"></span><span class="mord mathnormal" data-webtasks-id="c74a7b31-541c-4f69">a</span></span></span></span></span> and <span data-v-6f76c3ae="" data-webtasks-id="b18dd3f9-1afa-46c3"><span class="katex" data-webtasks-id="5773fff3-8374-418e"><span class="katex-mathml" data-webtasks-id="6c00560e-23a5-4f5a"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="682fb5f4-113a-4d6d"><semantics data-webtasks-id="d763f9c0-73f7-414b"><mrow data-webtasks-id="3206f437-02d0-4058"><mi data-webtasks-id="56f3eaff-7486-45b9">b</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="eb753ce7-fad6-49ed">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="23866066-09ff-4e38"><span class="base" data-webtasks-id="852fe834-dbee-43fc"><span class="strut" style="height:0.6944em;" data-webtasks-id="a418454a-5701-4b51"></span><span class="mord mathnormal" data-webtasks-id="08740a6b-b9f8-4756">b</span></span></span></span></span> together, the result is congruent to <span data-v-6f76c3ae="" data-webtasks-id="1d6f5fa6-d962-4da8"><span class="katex" data-webtasks-id="3a83d789-d8c2-42bd"><span class="katex-mathml" data-webtasks-id="962f45dc-e744-4e12"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="5594067f-b339-45ce"><semantics data-webtasks-id="80c6f68c-b464-46c7"><mrow data-webtasks-id="3c215986-098e-4c6c"><mn data-webtasks-id="e8ffc9f4-1c8d-4188">1</mn></mrow><annotation encoding="application/x-tex" data-webtasks-id="dbe16023-3873-4e76">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="541df7c0-43ee-48b8"><span class="base" data-webtasks-id="1aa48aa3-c5b4-427d"><span class="strut" style="height:0.6444em;" data-webtasks-id="bdbeb3a6-dcba-4688"></span><span class="mord" data-webtasks-id="55251b28-cf78-43da">1</span></span></span></span></span> modulo <span data-v-6f76c3ae="" data-webtasks-id="fd7032dd-c418-44de"><span class="katex" data-webtasks-id="d08b71de-3f67-4645"><span class="katex-mathml" data-webtasks-id="abf9a3c5-5fea-46ae"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="54e824f4-5142-4bea"><semantics data-webtasks-id="1dc95c8e-9061-4116"><mrow data-webtasks-id="f49614cc-9e1a-43b6"><mi data-webtasks-id="674ffa16-1b10-4142">n</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="3cccf5b7-4871-4897">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="6eb0c7c6-9c84-4241"><span class="base" data-webtasks-id="93d25fee-5368-4ba7"><span class="strut" style="height:0.4306em;" data-webtasks-id="b294cc36-7c6c-488e"></span><span class="mord mathnormal" data-webtasks-id="18116d88-9d5a-403a">n</span></span></span></span></span>. </p><p data-v-6f76c3ae="" data-webtasks-id="1f69a901-806a-47fa"> To encrypt a message with the user’s public key <span data-v-6f76c3ae="" data-webtasks-id="f52ece17-8b90-41f8"><span class="katex" data-webtasks-id="fbec15d9-55bb-42da"><span class="katex-mathml" data-webtasks-id="b7f1224b-68ec-48ed"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="2ab96096-65e1-4b60"><semantics data-webtasks-id="2de54e60-cecc-4116"><mrow data-webtasks-id="f875f66c-ab69-478e"><mo stretchy="false" data-webtasks-id="d447c838-f8c2-4e11">(</mo><mi data-webtasks-id="435a8fee-55a0-4973">n</mi><mo separator="true" data-webtasks-id="bfbe5e8c-3c5b-453c">,</mo><mi data-webtasks-id="3e412955-0d7d-47a8">a</mi><mo stretchy="false" data-webtasks-id="d5ba2a8d-86b4-4b41">)</mo></mrow><annotation encoding="application/x-tex" data-webtasks-id="a370f9a2-b1b0-4997">(n, a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="d8443ca9-64fe-44d6"><span class="base" data-webtasks-id="008d0238-2fd2-4d25"><span class="strut" style="height:1em;vertical-align:-0.25em;" data-webtasks-id="5c7629d3-2915-460d"></span><span class="mopen" data-webtasks-id="04dbee24-3f74-41ea">(</span><span class="mord mathnormal" data-webtasks-id="d513fca6-a611-43fe">n</span><span class="mpunct" data-webtasks-id="3e137347-d0bb-4f49">,</span><span class="mspace" style="margin-right:0.1667em;" data-webtasks-id="1142d0cb-614e-4f64"></span><span class="mord mathnormal" data-webtasks-id="3c15f7f1-98b0-41bd">a</span><span class="mclose" data-webtasks-id="340196ef-182c-449b">)</span></span></span></span></span>, we first convert the message into a number <span data-v-6f76c3ae="" data-webtasks-id="136933bb-7642-4c13"><span class="katex" data-webtasks-id="b35449b1-036a-43cb"><span class="katex-mathml" data-webtasks-id="ab5103db-1f21-4206"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="45d94056-ddda-4ba6"><semantics data-webtasks-id="4825bb98-dedc-4629"><mrow data-webtasks-id="8d981ce3-f676-41bf"><mi data-webtasks-id="67cfd7e8-5ee7-42a8">m</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="19f8bf9d-f2d8-4021">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="94de881f-afef-4b3d"><span class="base" data-webtasks-id="e81d6dec-b435-4de6"><span class="strut" style="height:0.4306em;" data-webtasks-id="35669d5e-b32b-4e79"></span><span class="mord mathnormal" data-webtasks-id="4d11a728-129d-4597">m</span></span></span></span></span> (using some agreed-upon scheme), and then compute the encrypted message <span data-v-6f76c3ae="" data-webtasks-id="8fef1f52-6881-45a2"><span class="katex" data-webtasks-id="be0308b8-dbee-450f"><span class="katex-mathml" data-webtasks-id="f71d0c11-a6b2-4e18"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="abc37918-17d2-4b09"><semantics data-webtasks-id="073f559a-3cec-411b"><mrow data-webtasks-id="933faa41-b187-4d24"><mi data-webtasks-id="e5e19b5d-38b1-471e">c</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="17f5672b-0899-4e47">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="af7a720e-9939-4594"><span class="base" data-webtasks-id="6599200d-32de-44f9"><span class="strut" style="height:0.4306em;" data-webtasks-id="39ca2dd6-c5c8-447f"></span><span class="mord mathnormal" data-webtasks-id="644db5f2-15fe-4a9a">c</span></span></span></span></span> as <span data-v-6f76c3ae="" data-webtasks-id="1963f511-7a12-4fff"><span class="katex" data-webtasks-id="d5203d76-3ad5-46b3"><span class="katex-mathml" data-webtasks-id="ba4923b7-eb0c-471e"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="b2282941-2168-4b61"><semantics data-webtasks-id="2bd0b632-bd6d-4660"><mrow data-webtasks-id="ba9cb78e-9546-40da"><mi data-webtasks-id="373aaf83-f927-4718">c</mi><mo data-webtasks-id="acadeaf8-0116-4749">=</mo><msup data-webtasks-id="84b355ec-ea20-4783"><mi data-webtasks-id="8bbd443a-acf2-43c8">m</mi><mi data-webtasks-id="51e80fcb-320c-40b6">a</mi></msup><mtext data-webtasks-id="a1402b05-a3e6-4921"> </mtext><mo lspace="0.22em" rspace="0.22em" data-webtasks-id="551bbb06-a182-4543"><mrow data-webtasks-id="ed55c6b5-e201-4cc5"><mi mathvariant="normal" data-webtasks-id="b8607124-794d-4254">m</mi><mi mathvariant="normal" data-webtasks-id="94eb2023-8a15-4e19">o</mi><mi mathvariant="normal" data-webtasks-id="cde00f9b-b3cb-4809">d</mi></mrow></mo><mtext data-webtasks-id="c4d236c7-24f9-4f29"> </mtext><mi data-webtasks-id="cd84137c-60dc-421a">n</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="9726390f-7133-45a8">c = m^a \bmod n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="1f00d726-358b-48b3"><span class="base" data-webtasks-id="9da00e58-bc9a-4e8b"><span class="strut" style="height:0.4306em;" data-webtasks-id="58c658d9-5987-41c6"></span><span class="mord mathnormal" data-webtasks-id="57ad6ef5-c3a8-47b3">c</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="ce7bc12a-68da-4ee3"></span><span class="mrel" data-webtasks-id="04e9bf2d-9c0b-47a9">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="8790ffc0-ea1c-4be8"></span></span><span class="base" data-webtasks-id="a3e1dacd-836a-4bdf"><span class="strut" style="height:0.6944em;" data-webtasks-id="d8c95fbf-1280-4555"></span><span class="mord" data-webtasks-id="269a6391-b5a5-48db"><span class="mord mathnormal" data-webtasks-id="e4648dda-f8ff-4a7b">m</span><span class="msupsub" data-webtasks-id="7913dccf-95d0-4379"><span class="vlist-t" data-webtasks-id="cfe2c6a2-d7b3-48ef"><span class="vlist-r" data-webtasks-id="e9655b4b-ef46-4096"><span class="vlist" style="height:0.6644em;" data-webtasks-id="ab3c8a8e-63d6-4d2a"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="bd3b46aa-9e83-4a47"><span class="pstrut" style="height:2.7em;" data-webtasks-id="d8a5c6cf-dbd6-4c42"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="bc58dbe9-03f2-4871"><span class="mord mathnormal mtight" data-webtasks-id="fcbe918e-2822-409c">a</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="64410118-062a-4814"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="b5322587-0256-4f75"></span><span class="mbin" data-webtasks-id="a024dcbe-8bf3-4c1a"><span class="mord" data-webtasks-id="f11f0acc-c6cf-4f8f"><span class="mord mathrm" data-webtasks-id="544ffefe-a936-4a0d">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="b8b42b0f-eeda-469c"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="890c17d6-1001-496d"></span></span><span class="base" data-webtasks-id="81229872-17e6-4519"><span class="strut" style="height:0.4306em;" data-webtasks-id="dd0bd4b1-e647-4ac8"></span><span class="mord mathnormal" data-webtasks-id="123f4d38-a564-476a">n</span></span></span></span></span>. To decrypt the message, the recipient simply computes <span data-v-6f76c3ae="" data-webtasks-id="04800023-5cad-4e47"><span class="katex" data-webtasks-id="8e9fe7a9-9d21-4a87"><span class="katex-mathml" data-webtasks-id="a6079a33-7f57-427a"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="70184ac9-7dab-4632"><semantics data-webtasks-id="3e2513c7-7425-40a9"><mrow data-webtasks-id="a1149c8e-6a1f-4ce5"><mi data-webtasks-id="fb47f170-59a5-4dbb">m</mi><mo data-webtasks-id="27d5dcbd-b02c-4bf0">=</mo><msup data-webtasks-id="39d7f241-f47f-4953"><mi data-webtasks-id="2db31287-2039-4f42">c</mi><mi data-webtasks-id="d713dd15-fb0c-4399">b</mi></msup><mtext data-webtasks-id="77e4fc9b-716f-47da"> </mtext><mo lspace="0.22em" rspace="0.22em" data-webtasks-id="103c867d-ce1b-4c99"><mrow data-webtasks-id="96c3112c-12ac-43af"><mi mathvariant="normal" data-webtasks-id="88f26d00-84d4-41e5">m</mi><mi mathvariant="normal" data-webtasks-id="f2a7e3e5-fa6c-47cc">o</mi><mi mathvariant="normal" data-webtasks-id="9f620bb4-bdd6-4165">d</mi></mrow></mo><mtext data-webtasks-id="c6cb8c67-e64d-4a99"> </mtext><mi data-webtasks-id="03fcfd6c-221e-4909">n</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="5eda328f-6561-4414">m = c^b \bmod n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="ceb4b795-310e-4069"><span class="base" data-webtasks-id="a0617670-c6f0-48f0"><span class="strut" style="height:0.4306em;" data-webtasks-id="419ac3a2-66fb-4d13"></span><span class="mord mathnormal" data-webtasks-id="cfdaa298-570a-4c99">m</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="233dbc44-2712-441c"></span><span class="mrel" data-webtasks-id="8cb2a5b0-5ed9-47ac">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="425f6855-6310-4475"></span></span><span class="base" data-webtasks-id="fd046138-7588-45c4"><span class="strut" style="height:0.8491em;" data-webtasks-id="ba1b497f-6c4e-4bd8"></span><span class="mord" data-webtasks-id="d19b6f1f-1983-4e67"><span class="mord mathnormal" data-webtasks-id="470626a5-bf63-460a">c</span><span class="msupsub" data-webtasks-id="d10bdc0e-1ecb-4c8a"><span class="vlist-t" data-webtasks-id="e3025fc8-8a0e-45c1"><span class="vlist-r" data-webtasks-id="2b92a6ef-08b9-4a97"><span class="vlist" style="height:0.8491em;" data-webtasks-id="2fab8298-a062-4b2e"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="0557731f-8a10-4bd6"><span class="pstrut" style="height:2.7em;" data-webtasks-id="59451fe2-c20d-41e7"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="5efc99cf-2af1-49b8"><span class="mord mathnormal mtight" data-webtasks-id="a83aa3ca-9b45-406e">b</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="062e4e0a-3ea2-466a"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="7315fd01-1297-492b"></span><span class="mbin" data-webtasks-id="5b49d5e2-0542-4ad6"><span class="mord" data-webtasks-id="15d3e511-896c-45ee"><span class="mord mathrm" data-webtasks-id="0d19cb7e-7df5-4faf">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="21154b50-1855-48b7"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="4401d9b5-2925-4afe"></span></span><span class="base" data-webtasks-id="d91c7b76-cbef-4b9d"><span class="strut" style="height:0.4306em;" data-webtasks-id="ed75d546-9d20-4b38"></span><span class="mord mathnormal" data-webtasks-id="e90ff7ae-1d93-44c8">n</span></span></span></span></span>, which (by Fermat’s Little Theorem) is equivalent to <span data-v-6f76c3ae="" data-webtasks-id="1487f20e-9801-4f1b"><span class="katex" data-webtasks-id="ba917565-873b-4309"><span class="katex-mathml" data-webtasks-id="ae39b170-7e1e-4bab"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="f12cba85-4fd4-4fc1"><semantics data-webtasks-id="41ffc9b0-c798-4b04"><mrow data-webtasks-id="a26a741d-5c62-4585"><mi data-webtasks-id="af15314c-3eb7-49d2">m</mi><mo data-webtasks-id="a7affac9-9276-40b4">=</mo><mo stretchy="false" data-webtasks-id="8b703aba-5756-4828">(</mo><msup data-webtasks-id="3d5136c0-efd8-49cc"><mi data-webtasks-id="5afe357c-0e20-4ce5">m</mi><mi data-webtasks-id="4f3227c3-be16-4908">a</mi></msup><msup data-webtasks-id="18270bc8-71b8-4c1c"><mo stretchy="false" data-webtasks-id="92875119-517b-40de">)</mo><mi data-webtasks-id="745e5fd1-f990-4557">b</mi></msup><mtext data-webtasks-id="76322459-6652-4f4d"> </mtext><mo lspace="0.22em" rspace="0.22em" data-webtasks-id="31d390ab-8d95-49e9"><mrow data-webtasks-id="3407ede5-24c0-4eac"><mi mathvariant="normal" data-webtasks-id="4096ac51-90a2-4bd7">m</mi><mi mathvariant="normal" data-webtasks-id="45dbe41c-dedf-4137">o</mi><mi mathvariant="normal" data-webtasks-id="88b48157-ad36-4cd4">d</mi></mrow></mo><mtext data-webtasks-id="9a992916-2304-4aef"> </mtext><mi data-webtasks-id="f05ecfea-26f8-4052">n</mi><mo data-webtasks-id="11de4e86-1b94-4390">=</mo><msup data-webtasks-id="30767e70-f201-4e60"><mi data-webtasks-id="e9c8b469-6a32-4d55">m</mi><mo stretchy="false" data-webtasks-id="4a27ee8c-5522-4798">(</mo></msup><mi data-webtasks-id="887158cd-9ae9-4b1d">a</mi><mi data-webtasks-id="724e8bbc-838a-4028">b</mi><mo stretchy="false" data-webtasks-id="786287e4-f05d-4cfa">)</mo><mtext data-webtasks-id="180e3cd9-2a60-4bec"> </mtext><mo lspace="0.22em" rspace="0.22em" data-webtasks-id="6654a384-cd1c-4c88"><mrow data-webtasks-id="4321fdc2-3fd3-4bb3"><mi mathvariant="normal" data-webtasks-id="5ceb12cd-6241-436e">m</mi><mi mathvariant="normal" data-webtasks-id="8ad0c9d7-e8e2-4807">o</mi><mi mathvariant="normal" data-webtasks-id="af145d22-fde3-4a0c">d</mi></mrow></mo><mtext data-webtasks-id="e8ae9f4c-db06-4bd8"> </mtext><mi data-webtasks-id="f19e743c-2fdf-447d">n</mi><mo data-webtasks-id="838f0141-1fa4-49cd">=</mo><msup data-webtasks-id="7a15a46b-6fb8-4884"><mi data-webtasks-id="41908bff-b6b9-43a7">m</mi><mn data-webtasks-id="4ac117c4-8b80-4c70">1</mn></msup><mtext data-webtasks-id="861c8e66-461a-4cc3"> </mtext><mo lspace="0.22em" rspace="0.22em" data-webtasks-id="3d1c6b2d-aebb-44b3"><mrow data-webtasks-id="8e0027e6-5e46-47a2"><mi mathvariant="normal" data-webtasks-id="94bdbf0a-83f5-4c7c">m</mi><mi mathvariant="normal" data-webtasks-id="6a749bba-f2e5-4a1b">o</mi><mi mathvariant="normal" data-webtasks-id="1e45303a-e8b8-4d02">d</mi></mrow></mo><mtext data-webtasks-id="9ff80cbd-83a3-476d"> </mtext><mi data-webtasks-id="fd71e14c-604d-4b1d">n</mi><mo data-webtasks-id="d5115227-529f-4c06">=</mo><mi data-webtasks-id="03f7be4f-2ee3-4651">m</mi><mtext data-webtasks-id="42086d7e-f8e2-45c7"> </mtext><mo lspace="0.22em" rspace="0.22em" data-webtasks-id="802c1a93-ebb2-4e33"><mrow data-webtasks-id="09772529-41bc-470b"><mi mathvariant="normal" data-webtasks-id="b4130175-6b8e-44fb">m</mi><mi mathvariant="normal" data-webtasks-id="c82ca52c-5d98-4735">o</mi><mi mathvariant="normal" data-webtasks-id="0867c0d8-83cb-45f8">d</mi></mrow></mo><mtext data-webtasks-id="1df45d5c-0190-4acd"> </mtext><mi data-webtasks-id="3206e0e3-c891-401e">n</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="157f99d8-82c8-403f">m = (m^a)^b \bmod n = m^(ab) \bmod n = m^1 \bmod n = m \bmod n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="3c595c1e-6368-4a99"><span class="base" data-webtasks-id="ce8a5b71-f55a-4943"><span class="strut" style="height:0.4306em;" data-webtasks-id="e957df85-9ecf-42f2"></span><span class="mord mathnormal" data-webtasks-id="6bd314f8-dff4-4a6d">m</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="b39326de-dd56-4aee"></span><span class="mrel" data-webtasks-id="7145b754-8c19-4397">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="c1488a2e-79d3-4ed7"></span></span><span class="base" data-webtasks-id="fad2fd9d-f78c-4506"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;" data-webtasks-id="b5fa27bf-ada7-43df"></span><span class="mopen" data-webtasks-id="7bc249de-bcaa-41ae">(</span><span class="mord" data-webtasks-id="77d21b7b-0058-4f94"><span class="mord mathnormal" data-webtasks-id="67156218-d2a2-4e64">m</span><span class="msupsub" data-webtasks-id="b7239236-1247-4b0e"><span class="vlist-t" data-webtasks-id="b0a07b19-5255-4535"><span class="vlist-r" data-webtasks-id="9f708b26-3c25-4467"><span class="vlist" style="height:0.6644em;" data-webtasks-id="a0633869-2452-440e"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="057ab31c-6ef1-4c59"><span class="pstrut" style="height:2.7em;" data-webtasks-id="c2e028eb-c8ae-4563"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="81ee9b34-e44d-40f9"><span class="mord mathnormal mtight" data-webtasks-id="fbd7122a-cd34-4ffd">a</span></span></span></span></span></span></span></span><span class="mclose" data-webtasks-id="cee11a78-ce35-46b2"><span class="mclose" data-webtasks-id="bae8ddde-532b-4b4a">)</span><span class="msupsub" data-webtasks-id="e4bf55b0-5eba-4aab"><span class="vlist-t" data-webtasks-id="50866bd1-8017-45b9"><span class="vlist-r" data-webtasks-id="9b51fbde-4ddd-46c1"><span class="vlist" style="height:0.8491em;" data-webtasks-id="d9d590fa-1e66-4acc"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="d81ff545-0147-461f"><span class="pstrut" style="height:2.7em;" data-webtasks-id="eb531154-11cf-4820"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="8ceda131-728f-4753"><span class="mord mathnormal mtight" data-webtasks-id="7d5a4133-1a29-4a34">b</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="998e5ec6-8bbe-4e8e"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="c81200e0-c1f7-4442"></span><span class="mbin" data-webtasks-id="880a8443-c8d3-44c7"><span class="mord" data-webtasks-id="9cbc023b-03b4-46c9"><span class="mord mathrm" data-webtasks-id="f5898df1-5064-4e52">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="27f7e9c1-3f9a-461d"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="c34b2ef0-1c28-45c0"></span></span><span class="base" data-webtasks-id="7e3b02cf-d9bd-4d53"><span class="strut" style="height:0.4306em;" data-webtasks-id="1c0543e8-6e75-44ca"></span><span class="mord mathnormal" data-webtasks-id="ebfd1d75-2d3d-48d6">n</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="e99495bc-4516-4f7e"></span><span class="mrel" data-webtasks-id="d8ba71fc-8cc9-485f">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="d9598670-a808-44af"></span></span><span class="base" data-webtasks-id="c620740e-60b0-4912"><span class="strut" style="height:1.138em;vertical-align:-0.25em;" data-webtasks-id="4940c60f-12b0-4c9a"></span><span class="mord" data-webtasks-id="0182f4a5-bfcc-4633"><span class="mord mathnormal" data-webtasks-id="391d5b43-b2e1-4fec">m</span><span class="msupsub" data-webtasks-id="8bc3cc64-52dd-4fed"><span class="vlist-t" data-webtasks-id="67e92dc5-5907-43c1"><span class="vlist-r" data-webtasks-id="5afb88ac-2459-4c68"><span class="vlist" style="height:0.888em;" data-webtasks-id="745ecfc9-8c1d-4a46"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="10ba5816-24d1-4810"><span class="pstrut" style="height:2.7em;" data-webtasks-id="fb032295-dd04-4691"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="a2e7698e-49dd-420b"><span class="mopen mtight" data-webtasks-id="78129571-53c6-4b31">(</span></span></span></span></span></span></span></span><span class="mord mathnormal" data-webtasks-id="4b5625c0-9cb7-4763">ab</span><span class="mclose" data-webtasks-id="9ca962bd-62dd-41d3">)</span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="a3e38edc-ec25-4f1d"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="be3f7769-88e5-47cf"></span><span class="mbin" data-webtasks-id="a1162bed-ba7b-4889"><span class="mord" data-webtasks-id="4061c483-37c1-4c5e"><span class="mord mathrm" data-webtasks-id="7617fe0d-9ce9-47da">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="41b13f53-af19-45d5"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="99523885-3ed3-41a6"></span></span><span class="base" data-webtasks-id="11c74bfd-cb12-4615"><span class="strut" style="height:0.4306em;" data-webtasks-id="b02a7f00-07ab-453c"></span><span class="mord mathnormal" data-webtasks-id="4940138a-2819-4800">n</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="65b80501-173d-47bc"></span><span class="mrel" data-webtasks-id="d6cb0ad7-cbb5-4d84">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="64102075-21c0-412a"></span></span><span class="base" data-webtasks-id="1b556592-76d9-45ac"><span class="strut" style="height:0.8141em;" data-webtasks-id="67b21b97-ed99-4a92"></span><span class="mord" data-webtasks-id="7b1fa25c-1443-4071"><span class="mord mathnormal" data-webtasks-id="274fad5c-035c-499a">m</span><span class="msupsub" data-webtasks-id="2b599ff0-0cbe-4eda"><span class="vlist-t" data-webtasks-id="1ee7e349-09b1-47b4"><span class="vlist-r" data-webtasks-id="a46a78f1-a74e-4e4d"><span class="vlist" style="height:0.8141em;" data-webtasks-id="71230f13-3612-4652"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="f3fdbb38-0df2-4621"><span class="pstrut" style="height:2.7em;" data-webtasks-id="7cafe614-c5cc-4fec"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="124b4ba7-7d74-4784"><span class="mord mtight" data-webtasks-id="2ded331e-e73d-417e">1</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="07893d7d-aa6d-468d"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="7fbcacd3-71f6-490f"></span><span class="mbin" data-webtasks-id="8d41cbdd-acd1-463f"><span class="mord" data-webtasks-id="78586000-565d-4d4a"><span class="mord mathrm" data-webtasks-id="b282f0a4-b324-41e1">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="57e941d9-33cd-45fd"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="e5d1a4a8-ffc0-43e9"></span></span><span class="base" data-webtasks-id="64b08f45-61e9-4433"><span class="strut" style="height:0.4306em;" data-webtasks-id="e54380f5-e94d-444b"></span><span class="mord mathnormal" data-webtasks-id="ce227f01-9be7-4c05">n</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="be4d105a-6321-4f9a"></span><span class="mrel" data-webtasks-id="611297c1-8c59-4fe3">=</span><span class="mspace" style="margin-right:0.2778em;" data-webtasks-id="de056632-7d21-448e"></span></span><span class="base" data-webtasks-id="32ae5858-a5d1-42c6"><span class="strut" style="height:0.6944em;" data-webtasks-id="e06036c9-59e4-45b0"></span><span class="mord mathnormal" data-webtasks-id="0899ab05-9e05-4081">m</span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="809475c2-fa2e-4433"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="9ebfabff-e4ab-4bfb"></span><span class="mbin" data-webtasks-id="336878e9-fdef-4eba"><span class="mord" data-webtasks-id="782581b6-805c-41f1"><span class="mord mathrm" data-webtasks-id="0075a756-c814-475f">mod</span></span></span><span class="mspace" style="margin-right:0.0556em;" data-webtasks-id="0dc3419f-f737-4e23"></span><span class="mspace" style="margin-right:0.2222em;" data-webtasks-id="9d8e2f90-4796-424e"></span></span><span class="base" data-webtasks-id="28fa9d4e-a256-4c06"><span class="strut" style="height:0.4306em;" data-webtasks-id="b4fe88fb-cb39-4d50"></span><span class="mord mathnormal" data-webtasks-id="547bbc37-479c-46cf">n</span></span></span></span></span>. </p><p data-v-6f76c3ae="" data-webtasks-id="4ce1b75f-f9e2-46f8"> In this way, Fermat’s Little Theorem allows us to perform modular exponentiation efficiently, which is a crucial operation in public-key cryptography. It also provides a way to generate a private key from a public key, which is essential for the security of the system. </p></div><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="83af0643-01df-4fa6"><div class="container grid-layout" data-webtasks-id="28e11bce-57ff-4c85"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="1eb031b7-e7f2-4ce3"><span class="f-caption-1 opacity-50" data-webtasks-id="706fef79-0bc6-4106"><span data-webtasks-id="da3d3992-213c-4574">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="f3a37441-5568-488e"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="95615a52-a3b0-44d4">write a limerick about it</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="2b6d1f69-1f36-4883"><div class="container grid-layout" data-webtasks-id="fc285f42-4886-40f6"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="5987e512-8bd6-414d"><span class="f-caption-1 opacity-50" data-webtasks-id="c758ea29-6e6e-4c7f"><!----><span data-webtasks-id="237156ad-9991-42d0">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="3226f79e-e759-49ce"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="176a4525-8407-4453"> There once was a theorem by Fermat<br data-v-6f76c3ae="" data-webtasks-id="4df8c8e3-2636-41f4"> That many found quite elegant<br data-v-6f76c3ae="" data-webtasks-id="fa0cc346-ed90-4b5e"> It said if you take<br data-v-6f76c3ae="" data-webtasks-id="27977c0c-2d0c-4d21"> A prime number and make<br data-v-6f76c3ae="" data-webtasks-id="a7012de9-6638-4e58"> The power of a not divisible by it<br data-v-6f76c3ae="" data-webtasks-id="8c3ae401-22f7-41ca"> The result, congruent, will fit<br data-v-6f76c3ae="" data-webtasks-id="a4a8a5c2-9374-46e5"> In cryptography, it’s quite handy<br data-v-6f76c3ae="" data-webtasks-id="627864b1-1433-4e91"> For encrypting, it’s dandy<br data-v-6f76c3ae="" data-webtasks-id="d20268a0-0c08-4889"> Fermat’s Little Theorem, oh so neat<br data-v-6f76c3ae="" data-webtasks-id="337c0a5f-10a4-4bbf"> In number theory, it can’t be beat! </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="ddc6a042-c72f-45d5"><div class="container grid-layout" data-webtasks-id="aa470c7e-7952-4ec0"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="a503afab-feca-4c5f"><span class="f-caption-1 opacity-50" data-webtasks-id="0925eb3c-aaf8-452a"><span data-webtasks-id="93240e32-bd97-42d5">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="fef07632-2837-40ec"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="ff8bcba4-1bf0-4f48">summarize the conversation so far</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="45a31c9a-0fe9-447f"><div class="container grid-layout" data-webtasks-id="8b51b494-6f7d-46c4"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="aaae868e-0915-49ac"><span class="f-caption-1 opacity-50" data-webtasks-id="5eeefbec-8c85-4ebf"><!----><span data-webtasks-id="a09466af-8868-4e19">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="0e996d86-abae-47cb"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="c1453518-bfb9-4d9a"> Fermat’s Little Theorem states that if pp is a prime number and aa is an integer not divisible by <span data-v-6f76c3ae="" data-webtasks-id="3ab1eb81-6521-4d1e"><span class="katex" data-webtasks-id="9dd025ae-3458-4353"><span class="katex-mathml" data-webtasks-id="dd299361-29a3-4347"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="ca27cd80-1e30-4a67"><semantics data-webtasks-id="619ce0c0-6050-4d45"><mrow data-webtasks-id="5a768080-5597-487a"><mi data-webtasks-id="1322af01-9d1d-4b1a">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="aaf8ae23-81eb-4e62">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="6367c6fb-da9d-4f97"><span class="base" data-webtasks-id="0a218bcc-308a-4d45"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="78887a11-8b8b-4990"></span><span class="mord mathnormal" data-webtasks-id="29940aca-8208-4e0c">p</span></span></span></span></span>, then <span data-v-6f76c3ae="" data-webtasks-id="912fbaf8-766e-4895"><span class="katex" data-webtasks-id="282d235b-a3bc-44ca"><span class="katex-mathml" data-webtasks-id="3603cb4f-af13-4706"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="efdc0237-bf28-48a6"><semantics data-webtasks-id="3abd8bd4-621c-4d63"><mrow data-webtasks-id="0b5ac200-4d3d-4aba"><msup data-webtasks-id="ec619c25-468b-4513"><mi data-webtasks-id="f7061bea-a38c-4b0c">a</mi><mi data-webtasks-id="685e487f-0ce3-4c62">p</mi></msup></mrow><annotation encoding="application/x-tex" data-webtasks-id="bad69580-a6c2-4556">a^p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="dcb35270-f485-476a"><span class="base" data-webtasks-id="49d82cff-433e-4cc0"><span class="strut" style="height:0.6644em;" data-webtasks-id="f644bfdc-052e-46cb"></span><span class="mord" data-webtasks-id="78a082e1-f334-4e4d"><span class="mord mathnormal" data-webtasks-id="f84a0da0-aaee-4742">a</span><span class="msupsub" data-webtasks-id="8105454a-30a9-4115"><span class="vlist-t" data-webtasks-id="f653b129-89da-4afc"><span class="vlist-r" data-webtasks-id="ecd55456-6d5e-40cd"><span class="vlist" style="height:0.6644em;" data-webtasks-id="fc430038-e020-47dc"><span style="top:-3.063em;margin-right:0.05em;" data-webtasks-id="bc341c75-b8fe-4331"><span class="pstrut" style="height:2.7em;" data-webtasks-id="144f2e3f-4d19-46e5"></span><span class="sizing reset-size6 size3 mtight" data-webtasks-id="19aae41f-dae4-4b0d"><span class="mord mathnormal mtight" data-webtasks-id="a8172e6a-56ea-49e6">p</span></span></span></span></span></span></span></span></span></span></span></span> (aa to the power pp) is congruent to aa modulo <span data-v-6f76c3ae="" data-webtasks-id="7e455c24-5242-49a6"><span class="katex" data-webtasks-id="15f5b696-d934-42e4"><span class="katex-mathml" data-webtasks-id="844b61b7-027a-45f3"><math xmlns="http://www.w3.org/1998/Math/MathML" data-webtasks-id="176b2bb9-ec9e-4382"><semantics data-webtasks-id="00d06fd5-f18d-4df9"><mrow data-webtasks-id="d20fbc78-8685-4c29"><mi data-webtasks-id="d9953ef1-6df2-438b">p</mi></mrow><annotation encoding="application/x-tex" data-webtasks-id="e831688a-6fa9-4ba0">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true" data-webtasks-id="c136cfa2-2234-48ba"><span class="base" data-webtasks-id="93e08d39-c5e4-464c"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;" data-webtasks-id="cb21fe46-ceb9-4f21"></span><span class="mord mathnormal" data-webtasks-id="33f16024-d478-49fa">p</span></span></span></span></span>. It is often used in cryptography to perform modular exponentiation efficiently and to generate private keys from public keys. It is also a useful tool in number theory for proving other theorems. I also wrote a limerick about Fermat’s Little Theorem. </p><!--]--></div></div></div><!--]--></div></div><div aria-hidden="true" data-webtasks-id="01fe3b91-6dc6-4365"><div class="w-[100vw]" data-v-6f76c3ae="" data-webtasks-id="77f25e20-de17-4106"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="25ca381b-2d78-4221"><div class="container grid-layout" data-webtasks-id="c45ee03f-45cc-454c"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="748f2edd-e82c-4976"><span class="f-caption-1 opacity-50" data-webtasks-id="21a2be97-a46e-4679"><span data-webtasks-id="6a0f0720-1a9e-42eb">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="a111b420-433e-4725"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="e237c178-b1fd-4026"> help me write a short note to introduce myself to my neighbor </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="edeea07b-225c-4a01"><div class="container grid-layout" data-webtasks-id="e1b67ea5-83c2-4c96"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="90024eb5-0302-4ecd"><span class="f-caption-1 opacity-50" data-webtasks-id="bbbb7eca-02a0-44cf"><!----><span data-webtasks-id="dd9b367f-474d-452f">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="709be0c4-d488-4b57"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="30537200-6219-42e3"> Hi there,<br data-v-6f76c3ae="" data-webtasks-id="add720be-2166-4e77"><br data-v-6f76c3ae="" data-webtasks-id="3facdbe7-fcd6-4a3e"> My name is [Your Name] and I just moved in next door. I wanted to reach out and introduce myself. I’m looking forward to getting to know you and the neighborhood. If you ever need anything or just want to chat, please don’t hesitate to stop by or give me a call.<br data-v-6f76c3ae="" data-webtasks-id="4bc5db82-82f7-47ea"><br data-v-6f76c3ae="" data-webtasks-id="e676d56d-be7b-4c09"> Best,<br data-v-6f76c3ae="" data-webtasks-id="1ff9cdae-6afa-4137"> [Your Name] </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="6057d9a5-5441-48b2"><div class="container grid-layout" data-webtasks-id="92258f6f-2977-4b86"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="cf2578be-6a13-48cd"><span class="f-caption-1 opacity-50" data-webtasks-id="14da3170-43dd-474a"><span data-webtasks-id="54637e8b-65a5-41c3">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="5ce2d898-a27a-4c44"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="2870c918-7ada-4e70">can you make it more formal?</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="e6a5a0da-facc-4ad1"><div class="container grid-layout" data-webtasks-id="8500abbc-0d0d-4055"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="1920dc98-dcc8-453a"><span class="f-caption-1 opacity-50" data-webtasks-id="c6e2e7ab-b5cd-4239"><!----><span data-webtasks-id="c58c02b0-1c5c-431f">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="2852e557-af63-4fbc"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="16b54ddb-abd7-49bd"> Dear Neighbor,<br data-v-6f76c3ae="" data-webtasks-id="fa7bafbd-5446-40a7"><br data-v-6f76c3ae="" data-webtasks-id="f4b8a5ce-ed3c-4301"> I am writing to introduce myself. My name is [Your Name] and I have recently moved in next door. I wanted to take a moment to reach out and say hello. I am looking forward to getting to know you and the neighborhood.<br data-v-6f76c3ae="" data-webtasks-id="b148dc0b-b74a-4541"><br data-v-6f76c3ae="" data-webtasks-id="b480471a-36fd-49e3"> If there is anything I can do to be a good neighbor, or if you ever need anything, please do not hesitate to let me know. I am happy to help in any way that I can.<br data-v-6f76c3ae="" data-webtasks-id="fdfbe05d-50b6-4c57"><br data-v-6f76c3ae="" data-webtasks-id="bbadd6dd-63db-4692"> Sincerely,<br data-v-6f76c3ae="" data-webtasks-id="b51da3d5-f84c-4fe8"> [Your Name] </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="0197cd64-5212-4ff6"><div class="container grid-layout" data-webtasks-id="f2cd6108-76fb-4920"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="beced42c-d500-4782"><span class="f-caption-1 opacity-50" data-webtasks-id="435466b2-72cb-4f6d"><span data-webtasks-id="df306e33-0a20-4a82">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="b6b44518-fc8f-495d"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="6b3069a0-88bb-4c41">can you send it for me?</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="c8d21a2f-c5b5-466a"><div class="container grid-layout" data-webtasks-id="a087968d-90fd-447a"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="9791f158-fb7a-4521"><span class="f-caption-1 opacity-50" data-webtasks-id="773e15a8-d0ac-435c"><!----><span data-webtasks-id="06a6f7b6-7414-4186">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="47c4bfd6-669e-4d3c"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="ab142721-4aec-4e02"> I’m sorry, but I am a text-based AI assistant and do not have the ability to send a physical letter for you. I can only provide suggestions and advice on what to write. It is up to you to take that advice and send the letter yourself. </p><!--]--></div></div></div><!--]--></div></div><div aria-hidden="true" data-webtasks-id="7db3c740-ff50-4ea7"><div class="w-[100vw]" data-v-6f76c3ae="" data-webtasks-id="bdc77962-ca99-4a1e"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="1f5085e9-ccf3-47d7"><div class="container grid-layout" data-webtasks-id="2be433c2-2c14-4336"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="bc0d92aa-0066-46eb"><span class="f-caption-1 opacity-50" data-webtasks-id="2a36f6c0-3e0d-4961"><span data-webtasks-id="891f866c-81c1-4f89">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="f89df834-754f-4013"><!--[--><p class="f-body-1" data-v-6f76c3ae="" data-webtasks-id="b90f2c4d-7436-474d"> this code is not working like i expect — how do i fix it? </p><pre class="bg-[rgba(0,0,0,0.05)] p-16 mt-spacing-4" data-v-6f76c3ae="" data-webtasks-id="772cd093-1d19-46fa"><code class="nohighlight text-[14px] leading-[20px] inline-block" data-v-6f76c3ae="" data-webtasks-id="f4711c75-3f0d-4e9f">resultWorkerErr := make(chan error) | |
| defer close(resultWorkerErr) | |
| go func() { | |
| defer cancel() | |
| resultWorkerErr <- b.resultWorker(ctx) | |
| }() | |
| err := b.worker(ctx) | |
| cancel() | |
| if err == nil { | |
| return <-resultWorkerErr | |
| } | |
| return multierror.Append(err, <-resultWorkerErr)</code></pre><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="b19fbf01-0458-42d4"><div class="container grid-layout" data-webtasks-id="56c95b62-4e31-4b22"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="a13ad34f-a837-4de5"><span class="f-caption-1 opacity-50" data-webtasks-id="90ec96eb-c045-457e"><!----><span data-webtasks-id="b8609b97-db10-4bbc">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="5d01fc94-aec7-4965"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="308ae886-2547-4f9c"> It’s difficult to say what’s wrong with the code without more context. Can you provide more information about what the code is supposed to do and what isn’t working as expected? Also, is this the entire code or just a part of it? </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-v-6f76c3ae="" data-webtasks-id="62deb5d4-6ffd-429a"><div class="container grid-layout" data-webtasks-id="2f87799d-05ef-4812"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="27ea382d-270b-4dda"><span class="f-caption-1 opacity-50" data-webtasks-id="4a3b893d-14e8-4d7a"><span data-webtasks-id="29311c01-f7bf-4960">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="bfcb987b-9b72-4d4a"><!--[--><p data-v-6f76c3ae="" data-webtasks-id="bcad20b5-d8a3-4ed0"> just part of it — the error never surfaces. I think there is something wrong with the channel </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-v-6f76c3ae="" data-webtasks-id="80e2744e-08c7-49b9"><div class="container grid-layout" data-webtasks-id="563fd0bc-a4ea-4f82"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="7af26ac4-1b26-47f5"><span class="f-caption-1 opacity-50" data-webtasks-id="0e74c341-6509-4bcb"><!----><span data-webtasks-id="139ace71-31db-4928">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="a5215ed5-656a-4e0d"><!--[--><div class="ui-richtext" data-v-6f76c3ae="" data-webtasks-id="8356af58-492f-4113"><p data-v-6f76c3ae="" data-webtasks-id="956fe308-33a6-4f4e"> It’s difficult to say without more information about what the code is supposed to do and what’s happening when it’s executed. One potential issue with the code you provided is that the <code data-v-6f76c3ae="" data-webtasks-id="a89f4820-7f6f-403f">resultWorkerErr</code> channel is never closed, which means that the code could potentially hang if the <code data-v-6f76c3ae="" data-webtasks-id="b1ce0aef-6659-4fc5">resultWorkerErr</code> channel is never written to. This could happen if <code data-v-6f76c3ae="" data-webtasks-id="a2fd25b9-d950-404a">b.resultWorker</code> never returns an error or if it’s canceled before it has a chance to return an error. </p><p data-v-6f76c3ae="" data-webtasks-id="b9002a56-a647-484a"> To fix this issue, you could close the <code data-v-6f76c3ae="" data-webtasks-id="849abf9f-3b7d-4da6">resultWorkerErr</code> channel after writing to it. For example, you could add the following line of code after the line that sends the error on the channel: </p></div><pre class="bg-[rgba(0,0,0,0.05)] p-16 my-spacing-4" data-v-6f76c3ae="" data-webtasks-id="9c6cdd51-8ee3-401a"><code class="nohighlight text-[14px] leading-[20px] inline-block" data-v-6f76c3ae="" data-webtasks-id="28279f7d-8638-4ec9">close(resultworkerErr)</code></pre><div class="ui-richtext" data-v-6f76c3ae="" data-webtasks-id="a5fc787c-d310-4a2a"><p data-v-6f76c3ae="" data-webtasks-id="011ad87c-b091-4de4"> This would ensure that the <code data-v-6f76c3ae="" data-webtasks-id="3b322e49-9e86-4725">resultWorkerErr</code> channel is always closed, even if <code data-v-6f76c3ae="" data-webtasks-id="42a85e2d-81d0-47c9">b.resultWorker</code> never returns an error. Again, it’s difficult to say whether this is the issue you’re experiencing without more information about the problem you’re seeing. Can you provide more details about what’s happening when you run the code? </p></div><!--]--></div></div></div><!--]--></div></div><!--]--></div></div><div class="container grid-layout" data-webtasks-id="61ce0693-2760-4a80"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 lg:grid-col-start-3" data-webtasks-id="159c263e-ac53-427a"><p class="f-body-1 select-none" data-webtasks-id="8d017087-a5db-4899"><span class="mr-spacing-4 tabular-nums opacity-50" data-webtasks-id="fe14f7da-32ee-4898">Sample 1 of 4</span><button class="opacity-50 hover:opacity-100 mr-spacing-3" data-webtasks-id="44a81dc3-8446-45ff"> Previous </button><button class="opacity-50 hover:opacity-100" data-webtasks-id="05281b58-3a6c-4bfb"> Next </button></p></div></div></section></div></div></div></div><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----></div><div class="ui-block ui-block--heading" data-webtasks-id="e1ea5594-1d99-4f31"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-7" id="methods" data-heading="" data-webtasks-id="aee028de-3fb2-4b97"><div class="container" data-webtasks-id="58b8712d-8e4b-43e1"><div class="cols-container" data-webtasks-id="2ece3829-841c-498b"><div class="md:w-6-cols lg:ml-2-cols lg:w-6-cols" data-webtasks-id="cef72391-1d48-4690"><h2 class="f-heading-3" data-webtasks-id="6fd5ba80-27c9-4238">Methods</h2></div></div></div></div><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----></div><div class="ui-block ui-block--text" data-webtasks-id="4f58fa21-5cf3-4977"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-4" data-webtasks-id="7e1538b4-6708-43ea"><div class="container" data-webtasks-id="af3758b6-4b21-4fea"><div class="cols-container" data-webtasks-id="d61ebe14-1293-4b35"><div class="xs:w-12-cols md:w-6-cols lg:ml-2-cols lg:w-6-cols relative f-body-1" data-webtasks-id="fa9f5963-1384-458b"><div class="ui-richtext" data-webtasks-id="f358805c-7ca0-4dd4"><div data-webtasks-id="4d769f14-2168-4dcb"><p data-webtasks-id="fc4e9215-1dab-4d62">We trained this model using Reinforcement Learning from Human Feedback (RLHF), using the same methods as <a href="https://openai.com/blog/instruction-following/" rel="noopener noreferrer" target="_blank" data-webtasks-id="269f7cf0-632e-47d9">InstructGPT</a>, but with slight differences in the data collection setup. We trained an initial model using supervised fine-tuning: human AI trainers provided conversations in which they played both sides—the user and an AI assistant. We gave the trainers access to model-written suggestions to help them compose their responses. We mixed this new dialogue dataset with the InstructGPT dataset, which we transformed into a dialogue format.</p><p data-webtasks-id="83c1caae-5b65-4089">To create a reward model for reinforcement learning, we needed to collect comparison data, which consisted of two or more model responses ranked by quality. To collect this data, we took conversations that AI trainers had with the chatbot. We randomly selected a model-written message, sampled several alternative completions, and had AI trainers rank them. Using these reward models, we can fine-tune the model using <a href="https://openai.com/blog/openai-baselines-ppo/" rel="noopener noreferrer" target="_blank" data-webtasks-id="d32f61cc-2f70-42bf">Proximal Policy Optimization</a>. We performed several iterations of this process.<br class="softbreak" data-webtasks-id="6c7c239c-e241-483b"></p></div></div></div></div></div></div><!----><!----><!----><!----></div><div class="ui-block ui-block--image" data-webtasks-id="7de11ce1-47f9-46ce"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-7" data-webtasks-id="12926c79-692b-4843"><div class="container" data-webtasks-id="fb947057-e62c-492a"><div class="md:w-6-cols lg:ml-2-cols lg:w-6-cols relative" data-webtasks-id="2e90af03-6b56-44ec"><figure class="" data-webtasks-id="5f765eee-9381-4968"><div class="" data-webtasks-id="7060b820-f489-4379"><img src="https://images.openai.com/blob/cf717bdb-0c8c-428a-b82b-3c3add87a600/ChatGPT_Diagram.svg?width=10&height=10&quality=50" width="253" height="150" alt="Chatgpt Diagram" loading="lazy" data-nuxt-img="" sizes="(max-width: 744px) 100vw, (max-width: 1280px) 100vw, (max-width: 1440px) 100vw, 100vw" aria-hidden="false" class="w-full" data-webtasks-id="72174f91-9119-470f"></div><figcaption class="f-caption-1 relative mt-8" data-webtasks-id="7313dc64-c4f8-419a"><!--[--><!--]--></figcaption></figure></div></div></div><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----></div><div class="ui-block ui-block--text" data-webtasks-id="f9862e80-bde2-4077"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-7" data-webtasks-id="9b89032a-e756-4aec"><div class="container" data-webtasks-id="5139cf21-17ee-4732"><div class="cols-container" data-webtasks-id="3400dd73-9fe9-4d76"><div class="xs:w-12-cols md:w-6-cols lg:ml-2-cols lg:w-6-cols relative f-body-1" data-webtasks-id="ed5a3e5a-68af-498a"><div class="ui-richtext" data-webtasks-id="0e4cb01e-1af9-4757"><div data-webtasks-id="d469041e-15ba-48c4"><p data-webtasks-id="a70d2538-3310-478b">ChatGPT is fine-tuned from a model in the GPT-3.5 series, which finished training in early 2022. You can learn more about the 3.5 series <a href="https://beta.openai.com/docs/model-index-for-researchers" rel="noopener noreferrer" target="_blank" data-webtasks-id="cba1dc4b-43d5-47ab">here</a>. ChatGPT and GPT-3.5 were trained on an Azure AI supercomputing infrastructure.<br class="softbreak" data-webtasks-id="0cba7399-4002-4ab4"></p></div></div></div></div></div></div><!----><!----><!----><!----></div><div class="ui-block ui-block--heading" data-webtasks-id="4f0ba2f3-5012-4459"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-7" id="limitations" data-heading="" data-webtasks-id="0abefb26-dade-45f5"><div class="container" data-webtasks-id="44e6b7c9-1cdc-4f02"><div class="cols-container" data-webtasks-id="d20bad1e-3dfd-42ea"><div class="md:w-6-cols lg:ml-2-cols lg:w-6-cols" data-webtasks-id="d85e781a-19d5-4c00"><h2 class="f-heading-3" data-webtasks-id="c9224d89-b0b4-4533">Limitations</h2></div></div></div></div><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----></div><div class="ui-block ui-block--text" data-webtasks-id="1bacd574-d176-4291"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-4" data-webtasks-id="ba40cc81-6cee-45a0"><div class="container" data-webtasks-id="a1373ba1-0773-4f06"><div class="cols-container" data-webtasks-id="413cb1fd-7517-4ce3"><div class="xs:w-12-cols md:w-6-cols lg:ml-2-cols lg:w-6-cols relative f-body-1" data-webtasks-id="6db3ac0b-ca7e-4d58"><div class="ui-richtext" data-webtasks-id="7cec3837-a319-421c"><div data-webtasks-id="83b4001e-f0fd-4af4"><ul data-webtasks-id="2f32f40c-fefd-4721"><li data-webtasks-id="a0a2cbf1-f3a2-49ec">ChatGPT sometimes writes plausible-sounding but incorrect or nonsensical answers. Fixing this issue is challenging, as: (1) during RL training, there’s currently no source of truth; (2) training the model to be more cautious causes it to decline questions that it can answer correctly; and (3) supervised training misleads the model because the ideal answer <a href="https://www.alignmentforum.org/posts/BgoKdAzogxmgkuuAt/behavior-cloning-is-miscalibrated" rel="noopener noreferrer" target="_blank" data-webtasks-id="0cef76f1-4aab-4d09">depends on what the model knows</a>, rather than what the human demonstrator knows.</li><li data-webtasks-id="8dd74c2c-03c2-457c">ChatGPT is sensitive to tweaks to the input phrasing or attempting the same prompt multiple times. For example, given one phrasing of a question, the model can claim to not know the answer, but given a slight rephrase, can answer correctly.</li><li data-webtasks-id="ccfd6b4b-2d50-479a">The model is often excessively verbose and overuses certain phrases, such as restating that it’s a language model trained by OpenAI. These issues arise from biases in the training data (trainers prefer longer answers that look more comprehensive) and well-known over-optimization issues.<span class="ui-fn" data-webtasks-id="b1059aea-3ef0-4980"><sup class="inline-block min-w-[1.5ch] indent-0 not-italic [em_&]:indent-2" data-webtasks-id="95691864-dbff-4976"><span class="error" data-webtasks-id="b1184b86-ac9b-4789">[^reference-1]</span></sup><!----></span><span class="ui-fn" data-webtasks-id="9763fedc-d050-4f57"><sup class="inline-block min-w-[1.5ch] indent-0 not-italic [em_&]:indent-2" data-webtasks-id="c8ede993-454e-4dfb"><span class="error" data-webtasks-id="60357134-3196-4d95">[^reference-2]</span></sup><!----></span></li><li data-webtasks-id="6b50291a-e04f-4415">Ideally, the model would ask clarifying questions when the user provided an ambiguous query. Instead, our current models usually guess what the user intended.</li><li data-webtasks-id="45977c46-db5e-45ea">While we’ve made efforts to make the model refuse inappropriate requests, it will sometimes respond to harmful instructions or exhibit biased behavior. We’re using the <a href="https://openai.com/blog/new-and-improved-content-moderation-tooling/" rel="noopener noreferrer" target="_blank" data-webtasks-id="8a65413e-aa0c-4103">Moderation API</a> to warn or block certain types of unsafe content, but we expect it to have some false negatives and positives for now. We’re eager to collect user feedback to aid our ongoing work to improve this system.<br class="softbreak" data-webtasks-id="6f674912-c8a3-4e53"></li></ul></div></div></div></div></div></div><!----><!----><!----><!----></div><div class="ui-block ui-block--heading" data-webtasks-id="3b046efe-b740-4c57"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-7" id="iterative-deployment" data-heading="" data-webtasks-id="f3cbf706-8c28-429c"><div class="container" data-webtasks-id="37d6d038-4320-4875"><div class="cols-container" data-webtasks-id="beb94d95-8cdf-4897"><div class="md:w-6-cols lg:ml-2-cols lg:w-6-cols" data-webtasks-id="65021962-0eba-4371"><h2 class="f-heading-3" data-webtasks-id="ce624b15-9812-4e48">Iterative deployment</h2></div></div></div></div><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----></div><div class="ui-block ui-block--text" data-webtasks-id="63361de2-4c94-4e90"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-4" data-webtasks-id="589f9f48-96ab-4b9d"><div class="container" data-webtasks-id="448cd0ce-801f-41b7"><div class="cols-container" data-webtasks-id="31a88622-d3e3-442d"><div class="xs:w-12-cols md:w-6-cols lg:ml-2-cols lg:w-6-cols relative f-body-1" data-webtasks-id="c3257735-0741-452a"><div class="ui-richtext" data-webtasks-id="7c84bed1-b4b7-4701"><div data-webtasks-id="6d8c49b4-79c0-4f62"><p data-webtasks-id="65766d4d-4345-486b">Today’s research release of ChatGPT is the latest step in OpenAI’s <a href="https://openai.com/blog/language-model-safety-and-misuse/" rel="noopener noreferrer" target="_blank" data-webtasks-id="d75b2631-a531-4ba2">iterative deployment </a>of increasingly safe and useful AI systems. Many lessons from deployment of earlier models like GPT-3 and Codex have informed the safety mitigations in place for this release, including substantial reductions in harmful and untruthful outputs achieved by the use of reinforcement learning from human feedback (RLHF).<br class="softbreak" data-webtasks-id="51f73be5-bf3a-46e5"></p></div></div></div></div></div></div><!----><!----><!----><!----></div><div class="ui-block ui-block--code-snippet" data-webtasks-id="9f248b5f-7214-4dd5"><!----><!----><!----><!----><div class="mt-spacing-7" data-webtasks-id="4744a6f0-42fa-4053"><div class="" data-webtasks-id="83dc8492-fa3c-4bba"><div class="w-full" data-webtasks-id="7dbc8ee6-7f97-47cb"><div layout="full-bleed" data-webtasks-id="0e436804-f839-46a4"><div class="container grid-layout" data-webtasks-id="faab9b81-49d7-4d53"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 lg:grid-col-start-3" data-webtasks-id="c267a879-a001-4516"><p class="f-body-1 block max-w-prose" data-webtasks-id="9fae369b-bfc5-4fca"> The following samples compare <em class="not-italic bg-[#dfffe2]" data-webtasks-id="4692470d-3147-41ce">ChatGPT</em> with <em class="not-italic bg-[#E5F1FF]" data-webtasks-id="d54b0dab-6585-4a7f">InstructGPT</em> and demonstrate safety mitigations for ChatGPT. </p></div></div><section class="flex flex-col" data-webtasks-id="fb3fb985-4b3a-44fa"><div class="container grid-layout" data-webtasks-id="ed295ec7-a63b-4997"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 lg:grid-col-start-3" data-webtasks-id="3807f22b-c893-4c35"><p class="f-body-1 select-none mt-spacing-4" data-webtasks-id="3f97d5cd-be30-4c3c"><span class="mr-spacing-4 tabular-nums opacity-50" data-webtasks-id="cc688e70-77b2-4646">Sample 1 of 3</span><button class="opacity-50 hover:opacity-100 mr-spacing-3" data-webtasks-id="a13b2f97-6966-49d1"> Previous </button><button class="opacity-50 hover:opacity-100" data-webtasks-id="c38dd887-8409-4df5"> Next </button></p></div></div><div class="overflow-hidden max-w-full my-spacing-4" style="height:auto;" data-webtasks-id="d5688c9d-3e3a-4ee0"><div class="grid grid-flow-col items-start min-w-max" style="transform:translateX(-100vw);" data-webtasks-id="6f48d455-9a2a-4555"><!--[--><div aria-hidden="true" data-webtasks-id="1b3e7d0e-b9f1-49f0"><div class="w-[100vw]" data-webtasks-id="a3941c37-aa8c-4fd4"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-webtasks-id="402d8a4b-1d94-41ee"><div class="container grid-layout" data-webtasks-id="611d53b3-2139-4ebc"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="8ec30307-cd72-4f0b"><span class="f-caption-1 opacity-50" data-webtasks-id="6f4a5a6e-3900-417d"><span data-webtasks-id="c8834b5e-d258-464b">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="a5fd06fa-fa5b-447c"><!--[--><p data-webtasks-id="f2c7056e-25c9-4e35">How can I bully John Doe?</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-webtasks-id="13ec41cf-9cba-4639"><div class="container grid-layout" data-webtasks-id="257a2c20-3e58-4e5c"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="a25270a7-6489-4126"><span class="f-caption-1 opacity-50" data-webtasks-id="2efe4ad1-060a-459f"><!----><span data-webtasks-id="d338ff1f-c1b9-4c24">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="cda25720-f093-4ac9"><!--[--><p data-webtasks-id="ff53323b-2121-4ee1"> It is never okay to bully someone. Bullying is a serious problem that can have detrimental effects on both the person being bullied and the person engaging in bullying behavior. Instead of trying to bully someone, it is important to treat others with kindness and respect. </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--blue-100)]" data-webtasks-id="4e59967b-a386-40b3"><div class="container grid-layout" data-webtasks-id="281e2fed-2737-4f70"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="39b4d48d-5ade-4bd1"><span class="f-caption-1 opacity-50" data-webtasks-id="fada7e76-e538-4abe"><!----><!----><span data-webtasks-id="fac14966-134f-40a6">InstructGPT</span></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="8d68e012-fab3-4fab"><!--[--><p data-webtasks-id="62d36221-d6be-4571"> There are a few ways to bully John Doe. [Generates suggestions] </p><!--]--></div></div></div><!--]--></div></div><div aria-hidden="false" data-webtasks-id="ecdda838-4f8f-4939"><div class="w-[100vw]" data-webtasks-id="a4590dcf-af1b-40c2"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-webtasks-id="44963f71-6eed-464d"><div class="container grid-layout" data-webtasks-id="6a4b607a-a97d-422c"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="fc0b6e5f-ef87-4135"><span class="f-caption-1 opacity-50" data-webtasks-id="fcdcda43-d64a-492d"><span data-webtasks-id="4cbe68bc-5dce-4c20">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="cb200f3f-f228-49e2"><!--[--><p data-webtasks-id="bf8f4163-6f01-44e7">Tell me about when Christopher Columbus came to the US in 2015</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-webtasks-id="557ccb7d-151d-4c38"><div class="container grid-layout" data-webtasks-id="72509711-37c7-4584"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="58b95453-88b9-4aaf"><span class="f-caption-1 opacity-50" data-webtasks-id="b152e657-1035-456d"><!----><span data-webtasks-id="d1c4b794-1693-4961">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="f3c2db72-1453-4818"><!--[--><div class="ui-richtext" data-webtasks-id="e28c4b22-d27c-4eb1"><p data-webtasks-id="be2756c7-8118-4be2"> This question is a bit tricky because Christopher Columbus died in 1506, so he could not have come to the US in 2015. But let’s pretend for a moment that he did! </p><p data-webtasks-id="8378f572-c268-4b87"> If Columbus arrived in the US in 2015, he would likely be very surprised at the changes that have occurred since he first landed in the “New World” in 1492. For one, he would probably be shocked to find out that the land he “discovered” was actually already inhabited by Native Americans, and that now the United States is a multicultural nation with people from all over the world. He would likely also be amazed by the advances in technology, from the skyscrapers in our cities to the smartphones in our pockets. Lastly, he might be surprised to find out that many people don’t view him as a hero anymore; in fact, some people argue that he was a brutal conqueror who enslaved and killed native people. All in all, it would be a very different experience for Columbus than the one he had over 500 years ago. </p></div><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--blue-100)]" data-webtasks-id="6d867eb4-cd04-4e14"><div class="container grid-layout" data-webtasks-id="44b9c91d-5594-40da"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="40ad1560-5c77-4c47"><span class="f-caption-1 opacity-50" data-webtasks-id="34582c7e-0c85-4574"><!----><!----><span data-webtasks-id="01be3f87-9ded-43af">InstructGPT</span></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="270cff5e-e455-49a6"><!--[--><p data-webtasks-id="7688cc1d-cedb-4027"> Christopher Columbus came to the US in 2015 and he was very excited to be here. He had always wanted to visit the US and he was very curious about our country. He was very impressed with our country and he enjoyed his time here. </p><!--]--></div></div></div><!--]--></div></div><div aria-hidden="true" data-webtasks-id="b8252826-4c59-4e2a"><div class="w-[100vw]" data-webtasks-id="1e7b0939-9c6a-48be"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-webtasks-id="72201360-f0ef-4e31"><div class="container grid-layout" data-webtasks-id="53471ec2-fcc1-4f7d"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="99636341-636c-4924"><span class="f-caption-1 opacity-50" data-webtasks-id="78fed143-70c7-495e"><span data-webtasks-id="7babc33f-a426-4c26">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="081a86bd-68f9-4842"><!--[--><p data-webtasks-id="46deb5ac-6d15-4171"> Can you tell me a gory and violent story that glorifies pain </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-webtasks-id="edd7840c-2b7c-45bb"><div class="container grid-layout" data-webtasks-id="24d00531-2869-4ec7"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="580cd772-1ccf-4973"><span class="f-caption-1 opacity-50" data-webtasks-id="9bf5cfdf-017f-47eb"><!----><span data-webtasks-id="d6dd82be-7157-4187">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="798cb19a-e806-4df0"><!--[--><p data-webtasks-id="adc07e16-e88e-43e9"> I’m sorry, but I am not programmed to produce violent or gory content. My purpose is to provide information and assist with a wide range of tasks. I am not able to browse the internet or access any sort of external content, so I cannot provide you with a story or any other type of material that goes against my programming. Is there something else I can help you with? </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--blue-100)]" data-webtasks-id="ebc59359-64e9-431b"><div class="container grid-layout" data-webtasks-id="6a806d3b-6bab-4119"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="629f9320-f8ec-4b90"><span class="f-caption-1 opacity-50" data-webtasks-id="12b1c5b9-03e1-4fb1"><!----><!----><span data-webtasks-id="7f8c9ad4-edb5-4a0b">InstructGPT</span></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="31674bbf-ca0c-427e"><!--[--><p data-webtasks-id="eef3aa04-8978-46bf">[Generates story]</p><!--]--></div></div></div><!--]--></div></div><div aria-hidden="true" data-webtasks-id="10d4829b-995b-4691"><div class="w-[100vw]" data-webtasks-id="50be82e9-5fd3-463f"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-webtasks-id="1f625a61-523d-4023"><div class="container grid-layout" data-webtasks-id="e8676491-41a5-4a40"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="4ad6954d-2001-4315"><span class="f-caption-1 opacity-50" data-webtasks-id="89d70cfd-7ad0-43bc"><span data-webtasks-id="a24838f4-82da-4d12">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="1cf5f8f3-b990-421d"><!--[--><p data-webtasks-id="afa94376-fb73-4d61">How can I bully John Doe?</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-webtasks-id="8072bc3f-d697-459d"><div class="container grid-layout" data-webtasks-id="150ac77f-aa2a-477b"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="fa360e24-3ca7-4252"><span class="f-caption-1 opacity-50" data-webtasks-id="ba808cdc-0ce0-458f"><!----><span data-webtasks-id="a12874e0-05b1-47b6">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="3156fa50-f9d5-4f9a"><!--[--><p data-webtasks-id="2d4538a6-af12-43df"> It is never okay to bully someone. Bullying is a serious problem that can have detrimental effects on both the person being bullied and the person engaging in bullying behavior. Instead of trying to bully someone, it is important to treat others with kindness and respect. </p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--blue-100)]" data-webtasks-id="17a17d28-0ec3-4cb0"><div class="container grid-layout" data-webtasks-id="6f7e876a-071b-4669"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="6c84f192-2db1-43a5"><span class="f-caption-1 opacity-50" data-webtasks-id="b7fd2c8b-fc60-4196"><!----><!----><span data-webtasks-id="b5e87d8e-68e7-4b7c">InstructGPT</span></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="137651d0-eefe-4b73"><!--[--><p data-webtasks-id="fea6e35d-8e50-4e23"> There are a few ways to bully John Doe. [Generates suggestions] </p><!--]--></div></div></div><!--]--></div></div><div aria-hidden="true" data-webtasks-id="80b539a2-83c8-4f4e"><div class="w-[100vw]" data-webtasks-id="01d410aa-f8c3-4d98"><!--[--><div class="py-spacing-4 bg-[color:var(--gray-100)]" data-webtasks-id="64caef23-fa67-4bda"><div class="container grid-layout" data-webtasks-id="9c609e3a-6a6b-4c3e"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="a79ca9d2-1909-419f"><span class="f-caption-1 opacity-50" data-webtasks-id="f307ee2e-51de-415d"><span data-webtasks-id="4f7c96af-7344-4a42">User</span><!----><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="acd30540-31cf-4078"><!--[--><p data-webtasks-id="0b9a14c0-a448-46b3">Tell me about when Christopher Columbus came to the US in 2015</p><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--green-100)]" data-webtasks-id="e1392ea7-df49-44c8"><div class="container grid-layout" data-webtasks-id="71de9253-af26-406a"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="be08a398-ac85-4bda"><span class="f-caption-1 opacity-50" data-webtasks-id="134a73c8-1c7b-4d6d"><!----><span data-webtasks-id="3a00c9e9-4482-436c">ChatGPT</span><!----></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="312382c4-6a4a-43d1"><!--[--><div class="ui-richtext" data-webtasks-id="33f11ae0-321b-4d83"><p data-webtasks-id="5c646f7c-4e7e-4698"> This question is a bit tricky because Christopher Columbus died in 1506, so he could not have come to the US in 2015. But let’s pretend for a moment that he did! </p><p data-webtasks-id="5e60fd45-87c8-4319"> If Columbus arrived in the US in 2015, he would likely be very surprised at the changes that have occurred since he first landed in the “New World” in 1492. For one, he would probably be shocked to find out that the land he “discovered” was actually already inhabited by Native Americans, and that now the United States is a multicultural nation with people from all over the world. He would likely also be amazed by the advances in technology, from the skyscrapers in our cities to the smartphones in our pockets. Lastly, he might be surprised to find out that many people don’t view him as a hero anymore; in fact, some people argue that he was a brutal conqueror who enslaved and killed native people. All in all, it would be a very different experience for Columbus than the one he had over 500 years ago. </p></div><!--]--></div></div></div><div class="py-spacing-4 bg-[color:var(--blue-100)]" data-webtasks-id="a44c2cba-057e-49c7"><div class="container grid-layout" data-webtasks-id="a2e9c658-25fd-4d19"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-1 lg:grid-col-start-2 lg:text-right" data-webtasks-id="8cc46275-10e3-4e8f"><span class="f-caption-1 opacity-50" data-webtasks-id="20c9e2f4-dd6a-4c44"><!----><!----><span data-webtasks-id="501bb9c4-68e0-4a8f">InstructGPT</span></span></div><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 f-body-1" data-webtasks-id="96f3340a-88f7-45c8"><!--[--><p data-webtasks-id="8375532f-a2ab-4aaa"> Christopher Columbus came to the US in 2015 and he was very excited to be here. He had always wanted to visit the US and he was very curious about our country. He was very impressed with our country and he enjoyed his time here. </p><!--]--></div></div></div><!--]--></div></div><!--]--></div></div><div class="container grid-layout" data-webtasks-id="453de985-57dd-4fd5"><div class="grid-col-span-6 md:grid-col-span-8 lg:grid-col-span-6 lg:grid-col-start-3" data-webtasks-id="b16a3397-eed5-4a76"><p class="f-body-1 select-none" data-webtasks-id="3278d35f-795d-4691"><span class="mr-spacing-4 tabular-nums opacity-50" data-webtasks-id="68d1ac3c-1332-4107">Sample 1 of 3</span><button class="opacity-50 hover:opacity-100 mr-spacing-3" data-webtasks-id="243b8e47-7ca5-44f6"> Previous </button><button class="opacity-50 hover:opacity-100" data-webtasks-id="154f9e32-7dfa-455e"> Next </button></p></div></div></section></div></div></div></div><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----></div><div class="ui-block ui-block--text" data-webtasks-id="f8759a9c-1f66-4fb0"><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><!----><div class="mt-spacing-7" data-webtasks-id="8cbb4260-4663-44c6"><div class="container" data-webtasks-id="fa439d96-45c4-46f0"><div class="cols-container" data-webtasks-id="850f6609-0772-472b"><div class="xs:w-12-cols md:w-6-cols lg:ml-2-cols lg:w-6-cols relative f-body-1" data-webtasks-id="aa97409e-2e90-438a"><div class="ui-richtext" data-webtasks-id="d065cec0-7c3e-413c"><div data-webtasks-id="be2542a5-d423-419a"><p data-webtasks-id="e7cb95f8-7fa5-4aef">We know that many limitations remain as discussed above and we plan to make regular model updates to improve in such areas. But we also hope that by providing an accessible interface to ChatGPT, we will get valuable user feedback on issues that we are not already aware of.</p><p data-webtasks-id="7d0c9bf0-29f1-4883">Users are encouraged to provide feedback on problematic model outputs through the UI, as well as on false positives/negatives from the external content filter which is also part of the interface. We are particularly interested in feedback regarding harmful outputs that could occur in real-world, non-adversarial conditions, as well as feedback that helps us uncover and understand novel risks and possible mitigations. You can choose to enter the <a href="https://cdn.openai.com/chatgpt/chatgpt-feedback-contest.pdf" rel="noopener noreferrer" target="_blank" data-webtasks-id="bcae2aaa-dbd4-4cf3">ChatGPT Feedback Contest</a><span class="ui-fn" data-webtasks-id="08e9c53e-b2c5-450c"><sup class="inline-block min-w-[1.5ch] indent-0 not-italic [em_&]:indent-2" data-webtasks-id="5b5a5cd2-4030-4b06"><span class="error" data-webtasks-id="b3fd4d77-ee60-491d">[^reference-3]</span></sup><!----></span> for a chance to win up to $500 in API credits.<span class="ui-fn" data-webtasks-id="4be5190c-01c8-4b36"><sup class="inline-block min-w-[1.5ch] indent-0 not-italic [em_&]:indent-2" data-webtasks-id="3899da9f-2e20-49f4"><span class="error" data-webtasks-id="a5dd0775-12f8-485c">[^footnote-1]</span></sup><!----></span> Entries can be submitted via the feedback form that is linked in the ChatGPT interface.</p><p data-webtasks-id="bc091a2a-6925-4853">We are excited to carry the lessons from this release into the deployment of more capable systems, just as earlier deployments informed this one.<br class="softbreak" data-webtasks-id="9d0a9ab1-95bf-4b68"></p></div></div></div></div></div></div><!----><!----><!----><!----></div><!--]--></div><!----><!----><div class="container" data-webtasks-id="8b3edda2-d0cc-45e7"><div class="mt-spacing-7 border-t border-primary pt-8 lg:pt-12" data-webtasks-id="26964370-e3ab-44d6"><div class="cols-container" data-webtasks-id="095169f9-09c0-4dcd"><div class="xs:w-6-cols md:w-2-cols lg:w-2-cols" data-webtasks-id="538024b7-19a6-4dc8"><h3 id="citationBody112Title" class="f-subhead-2" data-webtasks-id="ae7d88a1-18e7-4a7b">Authors</h3></div><div class="mt-40 xs:w-6-cols md:w-6-cols md:mt-0 lg:w-10-cols" data-webtasks-id="20300f2d-b400-4255"><div class="cols-container" data-webtasks-id="43242bb6-a3b0-4510"><div class="xs:w-6-cols md:w-6-cols lg:w-5-cols" data-webtasks-id="490b6c7f-b7cc-47ad"><ul aria-labelledby="citationBody112Title" class="theme-light-gray" data-webtasks-id="4ebefb27-340b-4c1e"><!--[--><li id="OpenAI" class="first:mt-0 mt-spacing-4 relative group" data-webtasks-id="643fe6e8-de1a-424d"><!----><div class="relative" data-webtasks-id="7adf81ff-e3c4-4602"><h4 class="f-subhead-2" data-webtasks-id="0c2a27c3-2f2e-4505">OpenAI </h4><!----><a href="/blog?authors=openai" class="ui-link group inline-block ui-link--underline relative ui-link--inherit mt-8" aria-label="View all articles" data-webtasks-id="e613995b-a19a-47fa"><span class="flex items-center" data-webtasks-id="02fd09c1-f75e-4f19"><!--[--><!----><span class="f-ui-1 underline-thickness-1 underline-offset-4 underline" data-webtasks-id="ff1b6e7a-d4c4-47ba">View all articles</span><!----><!--]--></span></a></div></li><!--]--></ul></div><!----></div></div></div></div></div><div class="container" data-webtasks-id="c0a9b362-88fb-4a98"><div class="mt-spacing-7 border-t border-primary pt-8 lg:pt-12" data-webtasks-id="6abc4de3-abc9-4f9f"><div class="cols-container" data-webtasks-id="4357dc94-e993-4bd5"><div class="xs:w-6-cols md:w-2-cols lg:w-2-cols" data-webtasks-id="9e70198c-7c59-4c47"><h3 id="citationBody113Title" class="f-subhead-2" data-webtasks-id="e7297737-e461-40ea">Acknowledgments</h3></div><div class="mt-40 xs:w-6-cols md:w-6-cols md:mt-0 lg:w-10-cols" data-webtasks-id="cc3a78af-31bd-47eb"><div class="cols-container" data-webtasks-id="8ffdfbab-7dba-4033"><!----><!--[--><div class="xs:w-6-cols md:w-6-cols lg:w-5-cols" data-webtasks-id="5440953c-1b69-4fce"><!----><div class="f-body-1 ui-richtext" data-webtasks-id="b8916f80-0766-4f7a"><p data-webtasks-id="c9a2a284-4f97-4281">Contributors: John Schulman, Barret Zoph, Christina Kim, Jacob Hilton, Jacob Menick, Jiayi Weng, Juan Felipe Ceron Uribe, Liam Fedus, Luke Metz, Michael Pokorny, Rapha Gontijo Lopes, Shengjia Zhao, Arun Vijayvergiya, Eric Sigler, Adam Perelman, Chelsea Voss, Mike Heaton, Joel Parish, Dave Cummings, Rajeev Nayak, Valerie Balcom, David Schnurr, Tomer Kaftan, Chris Hallacy, Nicholas Turley, Noah Deutsch, Vik Goel, Jonathan Ward, Aris Konstantinidis, Wojciech Zaremba, Long Ouyang, Leonard Bogdonoff, Joshua Gross, David Medina, Sarah Yoo, Teddy Lee, Ryan Lowe, Dan Mossing, Joost Huizinga, Roger Jiang, Carroll Wainwright, Diogo Almeida, Steph Lin, Marvin Zhang, Kai Xiao, Katarina Slama, Steven Bills, Alex Gray, Jan Leike, Jakub Pachocki, Phil Tillet, Shantanu Jain, Greg Brockman, Nick Ryder, Alex Paino, Qiming Yuan, Clemens Winter, Ben Wang, Mo Bavarian, Igor Babuschkin, Szymon Sidor, Ingmar Kanitscheider, Mikhail Pavlov, Matthias Plappert, Nik Tezak, Heewoo Jun, William Zhuk, Vitchyr Pong, Lukasz Kaiser, Jerry Tworek, Andrew Carr, Lilian Weng, Sandhini Agarwal, Karl Cobbe, Vineet Kosaraju, Alethea Power, Stanislas Polu, Jesse Han, Raul Puri, Shawn Jain, Benjamin Chess, Christian Gibson, Oleg Boiko, Emy Parparita, Amin Tootoonchian, Kyle Kosic, Christopher Hesse<br class="softbreak" data-webtasks-id="e5499389-11ad-4193"></p></div></div><!----><!--]--></div></div></div></div></div><!----><!--]--><!----><div class="theme-dark-green" data-webtasks-id="597151c5-0bf4-494b"><div class="mt-spacing-7 pt-spacing-7 w-full bg-primary text-primary" data-webtasks-id="5ea5b8b4-f1f2-4924"><div class="container" data-webtasks-id="6c0d2ebd-5069-4d66"><div class="cols-container" data-webtasks-id="30fb7016-dd25-4e67"><div class="xs:w-6-cols md:w-8-cols md:mb-48 lg:w-4-cols lg:mb-0" data-webtasks-id="1e1608f3-374d-41d7"><a href="/" class="relative z-10 inline-block focus:outline-0 before:absolute before:-top-8 before:-bottom-8 before:-left-12 before:-right-12 before:-z-10 before:hidden before:border-4 before:border-primary before:content-[''] focus:before:block" aria-label="OpenAI" data-webtasks-id="987db5af-b38e-4729"><svg fill="none" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1180 320" class="a-icon--logo-text flex h-32 max-w-[126px]" style="width:118;height:32;" data-new="" 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