Question: What is name of the function used for the largest integer not greater than the number in question?

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

Answer: the floor function

are prime for any natural number n. <hl> Here represents the floor function , i.e., largest integer not greater than the number in question. <hl> The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

are prime for any natural number n. Here represents <hl> the floor function <hl>, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

Here represents <hl> the floor function <hl> , i.e., largest integer not greater than the number in question.


Question: Who first proved Bertrand's postulate?

Answer: Chebyshev

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. <hl> The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev ), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. <hl> However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by <hl> Chebyshev <hl>), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

The latter formula can be shown using Bertrand's postulate (proven first by <hl> Chebyshev <hl> ), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3.


Question: For what size natural number does Bertrand's postulate hold?

Answer: any natural number n > 3

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. <hl> The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3 . <hl> However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for <hl> any natural number n > 3 <hl>. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for <hl> any natural number n > 3 <hl> .


Question: How is the prime number p in Bertrand's postulate expressed mathematically?

Answer: n < p < 2n − 2

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. <hl> The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2 , for any natural number n > 3. <hl> However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with <hl> n < p < 2n − 2 <hl>, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with <hl> n < p < 2n − 2 <hl> , for any natural number n > 3.


Question: On what theorem is the formula that frequently generates the number 2 and all other primes precisely once based on?

Answer: Wilson's theorem

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. <hl> Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. <hl>

are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on <hl> Wilson's theorem <hl> and generates the number 2 many times and all other primes exactly once.


Question: What is another way to state the condition that infinitely many primes can exist only if a and q are coprime?

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

Answer: their greatest common divisor is one

<hl> can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one . <hl> If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

can have infinitely many primes only when a and q are coprime, i.e., <hl> their greatest common divisor is one <hl>. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

can have infinitely many primes only when a and q are coprime, i.e., <hl> their greatest common divisor is one <hl> .


Question: If a and q are coprime, which theorem holds that an arithmetic progression has an infinite number of primes?

Answer: Dirichlet's theorem

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. <hl> If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. <hl> The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, <hl> Dirichlet's theorem <hl> on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.


Question: What is the density of all primes compatible with a modulo 9?

Answer: 1/6

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. <hl> What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6 . <hl>

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is <hl> 1/6 <hl>.

What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is <hl> 1/6 <hl> .


Question: If q=9 and a=3,6 or 9, how many primes would be in the progression?

Answer: at most one prime number

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. <hl> The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number . <hl> In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain <hl> at most one prime number <hl>. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

The rows (=progressions) starting with a = 3, 6, or 9 contain <hl> at most one prime number <hl> .


Question: If q=9 and a=1,2,4,5,7, or 8, how many primes would be in a progression?

Answer: infinitely many prime numbers

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. <hl> In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers . <hl> What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are <hl> infinitely many prime numbers <hl>. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

In all other rows (a = 1, 2, 4, 5, 7, and 8) there are <hl> infinitely many prime numbers <hl> .


Question: What function is related to prime numbers?

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,

Answer: The zeta function

<hl> The zeta function is closely related to prime numbers. <hl> For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,

<hl> The zeta function <hl> is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,


Question: What type of value would the zeta function have if there were finite primes?

Answer: a finite value

The zeta function is closely related to prime numbers. <hl> For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value . <hl> However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have <hl> a finite value <hl>. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,

For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have <hl> a finite value <hl> .


Question: What property of the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... shows that there is an infinite number of primes?

Answer: diverges

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... <hl> diverges (i.e., exceeds any given number), so there must be infinitely many primes. <hl> Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... <hl> diverges <hl> (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,


Question: What does it mean when a harmonic series diverges?

Answer: exceeds any given number

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. <hl> However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number ), so there must be infinitely many primes. <hl> Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., <hl> exceeds any given number <hl>), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,

However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., <hl> exceeds any given number <hl> ), so there must be infinitely many primes.


Question: Of what mathematical nature is the Basel problem?

Answer: identity

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. <hl> Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, <hl>

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following <hl> identity <hl> (Basel problem), due to Euler,


Question: When was the Riemann hypothesis proposed?

The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

Answer: 1859

<hl> The unproven Riemann hypothesis, dating from 1859 , states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. <hl> The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

The unproven Riemann hypothesis, dating from <hl> 1859 <hl>, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

The unproven Riemann hypothesis, dating from <hl> 1859 <hl> , states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2.


Question: According to the Riemann hypothesis, all zeroes of the ζ-function have real part equal to 1/2 except for what values of s?

Answer: s = −2, −4, ...,

<hl> The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. <hl> The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

The unproven Riemann hypothesis, dating from 1859, states that except for <hl> s = −2, −4, ..., <hl> all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.


Question: What does the Riemann hypothesis state the source of irregularity in the distribution of points comes from?

Answer: random noise

The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. <hl> The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise . <hl> From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from <hl> random noise <hl>. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from <hl> random noise <hl> .


Question: What type of prime distribution does the Riemann hypothesis propose is also true for short intervals near X?

Answer: asymptotic distribution

The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. <hl> From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). <hl> This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the <hl> asymptotic distribution <hl> of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.


Question: What type of prime distribution is characterized about x/log x of numbers less than x?

Answer: asymptotic distribution

The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. <hl> From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). <hl> This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the <hl> asymptotic distribution <hl> of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.


Question: Which conjecture holds that every even integer n greater than 2 can be expressed as a sum of two primes?

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

Answer: Goldbach's conjecture

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. <hl> One of them is Goldbach's conjecture , which asserts that every even integer n greater than 2 can be written as a sum of two primes. <hl> As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is <hl> Goldbach's conjecture <hl>, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

One of them is <hl> Goldbach's conjecture <hl> , which asserts that every even integer n greater than 2 can be written as a sum of two primes.


Question: When did Landau propose his four conjectural problems?

Answer: 1912

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. <hl> Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. <hl> One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from <hl> 1912 <hl> are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.


Question: As of February 2011, how many numbers has Goldbach's conjecture been proven to?

Answer: all numbers up to n = 2 · 1017

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. <hl> As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017 . <hl> Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for <hl> all numbers up to n = 2 · 1017 <hl>. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

As of February 2011[update], this conjecture has been verified for <hl> all numbers up to n = 2 · 1017 <hl> .


Question: Which theorem states that all large odd integers can be expressed as a sum of three primes?

Answer: Vinogradov's theorem

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. <hl> Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. <hl> Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example <hl> Vinogradov's theorem <hl> says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.


Question: Which theorem states that every large even integer can be written as a prime summed with a semiprime?

Answer: Chen's theorem

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. <hl> Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. <hl> Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. <hl> Chen's theorem <hl> says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.


Question: What conjecture holds that there is an infinite amount of twin primes?

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

Answer: twin prime conjecture

A third type of conjectures concerns aspects of the distribution of primes. <hl> It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 ( twin prime conjecture ). <hl> Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (<hl> twin prime conjecture <hl>). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 ( <hl> twin prime conjecture <hl> ).


Question: What is a twin prime?

Answer: pairs of primes with difference 2

A third type of conjectures concerns aspects of the distribution of primes. <hl> It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). <hl> Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, <hl> pairs of primes with difference 2 <hl> (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.


Question: Which conjecture holds that for any positive integer n, there is an infinite amount of pairs of consecutive primes differing by 2n?

Answer: Polignac's conjecture

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). <hl> Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. <hl> It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). <hl> Polignac's conjecture <hl> is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.


Question: Of what form is the infinite amount of primes that comprise the special cases of Schinzel's hypothesis?

Answer: n2 + 1

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. <hl> It is conjectured there are infinitely many primes of the form n2 + 1 . <hl> These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form <hl> n2 + 1 <hl>. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

It is conjectured there are infinitely many primes of the form <hl> n2 + 1 <hl> .


Question: What conjecture holds that there are always a minimum of 4 primes between the squares of consecutive primes greater than 2?

Answer: Brocard's conjecture

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. <hl> Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. <hl> Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. <hl> Brocard's conjecture <hl> says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.


Question: Besides the study of prime numbers, what general theory was considered the official example of pure mathematics?

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.

Answer: number theory

<hl> For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. <hl> In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.

For a long time, <hl> number theory <hl> in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.


Question: What British mathematician took pride in doing work that he felt had no military benefit?

Answer: G. H. Hardy

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. <hl> In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. <hl> However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician <hl> G. H. Hardy <hl> prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.


Question: When was it discovered that prime numbers could applied to the creation of public key cryptography algorithms?

Answer: the 1970s

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. <hl> However, this vision was shattered in the 1970s , when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. <hl> Prime numbers are also used for hash tables and pseudorandom number generators.

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in <hl> the 1970s <hl>, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.

However, this vision was shattered in <hl> the 1970s <hl> , when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms.


Question: Besides public key cryptography, what is another application for prime numbers?

Answer: hash tables

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. <hl> Prime numbers are also used for hash tables and pseudorandom number generators. <hl>

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for <hl> hash tables <hl> and pseudorandom number generators.


Question: What type of number generators make use of prime numbers?

Answer: pseudorandom number generators

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. <hl> Prime numbers are also used for hash tables and pseudorandom number generators . <hl>

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and <hl> pseudorandom number generators <hl>.

Prime numbers are also used for hash tables and <hl> pseudorandom number generators <hl> .


Question: Assuming p is a prime other than 2 or 5, then, according to Fermat's theorem, what type of decimal will 1/p always be?

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

Answer: a recurring decimal

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. <hl> Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal , whose period is p − 1 or a divisor of p − 1. <hl> The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always <hl> a recurring decimal <hl>, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always <hl> a recurring decimal <hl> , whose period is p − 1 or a divisor of p − 1.


Question: According to Fermat's theorem, what period does 1/p always have assuming p is prime that is not 2 or 5?

Answer: p − 1

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. <hl> Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. <hl> The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is <hl> p − 1 <hl> or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.


Question: According to Wilson's theorem, what factorial must be divisible by p if some integer p > 1 is to be considered prime?

Answer: (p − 1)! + 1

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. <hl> The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. <hl> Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial <hl> (p − 1)! + 1 <hl> is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.


Question: According to Wilson's theorem, what factorial must be divisible by n if some integer n > 4 is to be considered composite?

Answer: (n − 1)!

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. <hl> Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. <hl>

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if <hl> (n − 1)! <hl> is divisible by n.


Question: What condition what must be satisfied in order for 1/p to be expressed in base q instead of base 10 and still have a period of p - 1?

Answer: p is not a prime factor of q

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. <hl> The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q . <hl> Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that <hl> p is not a prime factor of q <hl>. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that <hl> p is not a prime factor of q <hl> .


Question: What is one type of public key cryptography algorithm?

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Answer: RSA

<hl> Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). <hl> RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Several public-key cryptography algorithms, such as <hl> RSA <hl> and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.


Question: What is another type of public key cryptography algorithm?

Answer: the Diffie–Hellman key exchange

<hl> Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange , are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). <hl> RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Several public-key cryptography algorithms, such as RSA and <hl> the Diffie–Hellman key exchange <hl>, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Several public-key cryptography algorithms, such as RSA and <hl> the Diffie–Hellman key exchange <hl> , are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.).


Question: How many bits are often in the primes used for RSA public key cryptography algorithms?

Answer: 512-bit

<hl> Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). <hl> RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, <hl> 512-bit <hl> primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.


Question: On what type of exponentiation does the Diffie–Hellman key exchange depend on?

Answer: modular exponentiation

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. <hl> The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation , while the reverse operation the discrete logarithm is thought to be a hard problem. <hl>

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for <hl> modular exponentiation <hl>, while the reverse operation the discrete logarithm is thought to be a hard problem.

The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for <hl> modular exponentiation <hl> , while the reverse operation the discrete logarithm is thought to be a hard problem.


Question: How many bits are typically used in the primes for the Diffie–Hellman key exchange?

Answer: 1024-bit

<hl> Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). <hl> RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and <hl> 1024-bit <hl> primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.


Question: What type of insect employs the use of prime numbers in its evolutionary strategy?

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

Answer: cicadas

<hl> The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. <hl> These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

The evolutionary strategy used by <hl> cicadas <hl> of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.


Question: Where do cicadas spend the majority of their lives?

Answer: as grubs underground

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. <hl> These insects spend most of their lives as grubs underground . <hl> They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives <hl> as grubs underground <hl>. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

These insects spend most of their lives <hl> as grubs underground <hl> .


Question: Other than 7 and 13, what other year interval do cicadas pupate?

Answer: 17 years

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. <hl> They only pupate and then emerge from their burrows after 7, 13 or 17 years , at which point they fly about, breed, and then die after a few weeks at most. <hl> The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or <hl> 17 years <hl>, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

They only pupate and then emerge from their burrows after 7, 13 or <hl> 17 years <hl> , at which point they fly about, breed, and then die after a few weeks at most.


Question: What is the logic behind the cicadas prime number evolutionary strategy?

Answer: make it very difficult for predators to evolve that could specialize as predators

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. <hl> The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. <hl> If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences <hl> make it very difficult for predators to evolve that could specialize as predators <hl> on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.


Question: How much larger would cicada predator populations be if cicada outbreaks occurred at 14 and 15 year intervals?

Answer: up to 2% higher

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. <hl> Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. <hl> Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be <hl> up to 2% higher <hl> than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.


Question: What does the word prime generally suggest?

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type.

Answer: indecomposability

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. <hl> Generally, "prime" indicates minimality or indecomposability , in an appropriate sense. <hl> For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type.

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or <hl> indecomposability <hl>, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type.

Generally, "prime" indicates minimality or <hl> indecomposability <hl> , in an appropriate sense.


Question: For a field F containing 0 and 1, what would be the prime field?

Answer: the smallest subfield

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. <hl> For example, the prime field is the smallest subfield of a field F containing both 0 and 1. <hl> It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type.

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is <hl> the smallest subfield <hl> of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type.


Question: How can any knot be distinctively indicated?

Answer: as a connected sum of prime knots

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. <hl> Any knot can be uniquely expressed as a connected sum of prime knots . <hl> Prime models and prime 3-manifolds are other examples of this type.

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed <hl> as a connected sum of prime knots <hl>. Prime models and prime 3-manifolds are other examples of this type.

Any knot can be uniquely expressed <hl> as a connected sum of prime knots <hl> .