Prime number

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In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. There exists an infinitude of prime numbers, as demonstrated by Euclid in about 300 B.C.. The first 30 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, and 113 (sequence in OEIS); see the list of prime numbers for a longer list.

The property of being a prime is called primality, and the word prime is also used as an adjective. Since 2 is the only even prime number, the term odd prime refers to all prime numbers greater than 2.

The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers. Prime numbers have been the subject of intense research, yet some fundamental questions remain such as the Riemann hypothesis or the Goldbach conjecture, which have been open for more than a century. The problem of modeling the distribution of prime numbers is a popular subject of investigation for number theorists: When looking at individual numbers, the primes seem to be randomly distributed, but the "global" distribution of primes follows well-defined laws.

The notion of prime number has been generalized in many different branches of mathematics. In the context of ring theory, a branch of abstract algebra, the term "prime element" has a specific meaning. Here, a ring element a is defined to be prime if whenever a divides b c for ring elements b and c, then a divides at least one of b or c. With this meaning, the additive inverse of any prime number is also prime. In other words, when considering the set of integers [\mathbb] as a ring, [-7] is a prime element. Without further specification, however, "prime number" always means a positive integer prime. Among rings of complex algebraic integers, Eisenstein primes and Gaussian primes may also be of interest.

History of prime numbers

Ancient Greeks such as Pythagoras are believed to have been the first to study prime numbers, and the simple sieve method for finding them is attributed to Eratosthenes.

Until the 19th century most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is still valid despite labelling 1 a prime, such as the work of Stern and Zeisel. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. The change in label occurred so that it can be said 'each number has a unique factorization into primes.' Gowers, 2002, p. 118: "The seemingly arbitrary exclusion of 1 from the definition of a prime ... does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes." See also [Arguments for and against the primality of 1].

For a long time, prime numbers were thought as having no possible application outside of number theory; this changed in the 1970s when the concepts of public-key cryptography were invented, in which prime numbers formed the basis of the first algorithms such as the RSA cryptosystem or the Diffie-Hellman key-exchange algorithm.

Prime divisors

The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of primes in a unique way, i.e. unique except for the order. Primes are thus the "basic building blocks" of the natural numbers. For example, we can write

[23244 = 2^2 \times 3 \times 13 \times 149]

and any other factorization of 23244 as the product of primes will be identical except for the order of the factors. See prime factorization algorithm for details for how to do this in practice for larger numbers.

The importance of this theorem is one of the reasons for the exclusion of 1 from the set of prime numbers. If 1 were admitted as a prime, the precise statement of the theorem would require additional qualifications.

Properties of primes

• All prime numbers end in 1, 3, 7, or 9, except for 2 and 5. (Numbers ending in 0, 2, 4, 6 or 8 can be divided by 2 and numbers ending in 5 can be divided by 5)

• If p is a prime number and p divides a product ab of integers, then p divides a or p divides b. This proposition was proved by Euclid and is known as Euclid's lemma. It is used in some proofs of the uniqueness of prime factorizations.

• The ring Z/nZ (see modular arithmetic) is a field if and only if n is a prime. Put another way: n is prime if and only if φ(n) = n − 1.

• If p is prime and a is any integer, then a^p − a is divisible by p (Fermat's little theorem).

• If p is a prime number other than 2 and 5, 1/p is always a recurring decimal, with a period of p-1 or a divisor of p-1. This can be deduced directly from Fermat's little theorem. 1/p expressed likewise in base q (i.e. other than base 10) has similar effect, provided that p is not a prime factor of q. The article on recurring decimals shows some of the interesting properties.

• An integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p (Wilson's theorem). Conversely, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

• If n is a positive integer greater than 1, then there is always a prime number p with n < p < 2n (Bertrand's postulate).

• Adding the reciprocals of all primes together results in a divergent infinite series (proof). More precisely, if S(x) denotes the sum of the reciprocals of all prime numbers p with p ≤ x, then S(x) = ln ln x + O(1) for x → ∞ (see Big O notation).

• In every arithmetic progression a, a + q, a + 2q, a + 3q,... where the positive integers a and q ≥ 1 are coprime, there are infinitely many primes (Dirichlet's theorem).

• The characteristic of every field is either zero or a prime number.

• If G is a finite group and pⁿ is the highest power of the prime p which divides the order of G, then G has a subgroup of order pⁿ. (Sylow theorems)

• If p is prime and G is a group with pⁿ elements, then G contains an element of order p.

• The prime number theorem says that the proportion of primes less than x is asymptotic to 1/ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x).

• The Copeland-Erdős constant 0.235711131719232931374143..., obtained by concatenating the prime numbers, is known to be an irrational number.

• The value of the Riemann zeta function at each point in the complex plane is given by a product over the set of all primes.

:[\zeta(s)=\sum_^\infin \frac = \prod_ \frac}]

Evaluating this identity at different integers provides an infinite number of products over the primes whose values can be calculated, the first two being

:[\prod_ \frac} = \infty]

:[\prod_ \frac}= \frac]

• If [p>1\;], the polynomial [ x^+x^+ \cdots + 1 ] is irreducible if and only if [p\;] is prime.

The number of prime numbers

There are infinitely many prime numbers

The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:

Suppose you have a finite number of primes. Call this number m. Multiply all m primes together and add one (see Euclid number). The resulting number is not divisible by any of the finite set of primes, because dividing by any of these would give a remainder of one. And one is not divisible by any primes. Therefore it must either be prime itself, or be divisible by some other prime that was not included in the finite set. Either way, there must be at least m + 1 primes. But this argument applies no matter what m is; it applies to m + 1, too. So there are more primes than any given finite number.

This previous argument explains why the product of m primes + 1 must be divisible by some prime not in the finite set of primes.

Other mathematicians have given their own proofs. One of those (due to Euler) shows that the sum of the reciprocals of all prime numbers diverges to infinity. Kummer's is particularly elegant and Harry Furstenberg provides one using general topology.Furstenberg, Harry. [On the infinitude of primes]. Amer. Math. Monthly 62, (1955), p. 353. A description of the proof is also available at http://www.everything2.com/index.pl?node_id=1460203

Counting the number of prime numbers below a given number

Even though the total number of primes is infinite, one could still ask "approximately how many primes are there below 100,000" or "How likely is a random 100-digit number to be prime?"

Provided one is not asking this of too large a value, one can use the prime counting function π(x) to find out exactly how many primes there are less than x. Values of x as large as 10²⁰ can be calculated quickly and accurately with modern computers. Thus, e.g., π(100000) = 9592.

For larger values of x beyond the reach of modern equipment, the prime number theorem provides a good heuristic estimate.

Location of prime numbers

Finding prime numbers

The Sieve of Eratosthenes is a simple way, and the Sieve of Atkin a fast way, to compute the list of all prime numbers up to a given limit.

In practice, though, one usually wants to check whether a given number is prime, rather than generate a list of primes. Further, it is often satisfactory to know the answer with a high probability. It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests. These typically pick a random number called a "witness" and check some formula involving the witness and the potential prime N. After several iterations, they declare N to be "definitely composite" or "probably prime". Some of these tests are not perfect: there may be some composite numbers, called pseudoprimes for the respective test, that will be declared "probably prime" no matter what witness is chosen. However, the most popular probabilistic tests do not suffer from this drawback.

One method for determining whether a number is prime is to divide by all primes less than or equal to the square root of that number. If any of the divisions come out as an integer, then the original number is not a prime. Otherwise, it is a prime. One need not actually calculate the square root; once one sees that the quotients exceed the divisors, one can stop. This is known as trial division; it is the simplest primality test and it quickly becomes impractical for testing large integers because the number of possible factors grows exponentially as the number of digits in the number-to-be-tested increases.

Pseudocode for programs to find primes

The least efficient algorithm starts out with a list containing just the number 2, then tries dividing each subsequent number by the primes already in the list. If it is not divisible evenly by any of them, it is added to the list. But if it is divisible evenly by any number already on the list, the program moves on to the next candidate.

The simplest, most elegant algorithm is perhaps the sieve of Eratosthenes.

limit = 1000000; // arbitrary search limit
for (i = 2; i <= limit; i++) 
for (n = 2; n <= sqrt (limit); n++) 
}

} for (n = 2; n =< limit; n++)

A more complicated, but more efficient algorithm (when properly optimized) is the sieve of Atkin. Its basic structure is as follows.

limit = 1000000 // arbitrary search limit
for (n = 5; n <= limit; n++) 


// put in candidate primes: 
// integers which have an odd number of representations by certain quadratic forms
for (x = 1; x <= sqrt (limit); x++) 

}

// eliminate composites by sieving for (n = 5; n <= sqrt (limit); n++) } }

print 2, 3 for (n = 5; n <= limit; n++)

Primality tests

Main article: primality test

A primality test algorithm is an algorithm which tests a number for primality, i.e. whether the number is a prime number.

• AKS primality test
• Fermat primality test
• Lucas-Lehmer test
• Solovay-Strassen primality test
• Miller-Rabin primality test

A probable prime is an integer which, by virtue of having passed a certain test, is considered to be probably prime. Probable primes which are in fact composite (such as Carmichael numbers) are called pseudoprimes.

In 2002, Indian scientists at IIT Kanpur discovered a new deterministic algorithm known as the AKS algorithm. The amount of time that this algorithm takes to check whether a number N is prime depends on a polynomial function of the number of digits of N (i.e. of the logarithm of N).

Formulas yielding prime numbers

There is no known formula for primes which is more efficient at finding primes than the methods mentioned above under "Finding prime numbers".

There is a set of Diophantine equations in 9 variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its positive values are prime.

There is no polynomial, even in several variables, that takes only prime values. For example, the curious polynomial in one variable f(n) = n² − n + 41 yields primes for n = 0,..., 40, but f(41) is composite. However, there are polynomials in several variables, whose positive values as the variables take all positive integer values are exactly the primes.

Another formula is based on Wilson's theorem mentioned above, and generates the number two many times and all other primes exactly once. There are other similar formulae which also produce primes.

Special types of primes from formulas for primes

A prime p is called primorial or prime-factorial if it has the form p = Π(n) ± 1 for some number n, where Π(n) stands for the product 2 · 3 · 5 · 7 · 11 · ... of all the primes ≤ n. A prime is called factorial if it is of the form n! ± 1. The first factorial primes are:

n! − 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,... (sequence in OEIS)

n! + 1 is prime for n = 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154... (sequence in OEIS)

The largest known primorial prime is Π(392113) + 1, found by Heuer in 2001.[link] The largest known factorial prime is 34790! − 1, found by Marchal, Carmody and Kuosa in 2002.[link] It is not known whether there are infinitely many primorial or factorial primes.

Primes of the form 2ⁿ − 1 are known as Mersenne primes, while primes of the form [2^ + 1] are known as Fermat primes. Prime numbers p where 2p + 1 is also prime are known as Sophie Germain primes. The following list is of other special types of prime numbers that come from formulas:

• Wieferich primes,
• Wilson primes,
• Wall-Sun-Sun primes,
• Wolstenholme primes,
• Unique primes,
• Newman-Shanks-Williams primes (NSW primes),
• Smarandache-Wellin primes,
• Wagstaff primes and
• Supersingular primes.

Some primes are classified according to the properties of their digits in decimal or other bases. For example, numbers whose digits form a palindromic sequence are called palindromic primes, and a prime number is called a truncatable prime if successively removing the first digit at the left or the right yields only new prime numbers.

The distribution of the prime numbers

The prime numbers are distributed among the natural numbers in an unpredictable way, but there do appear to be laws governing their behavior. Leonhard Euler commented

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate. (Havil 2003, p. 163)

Paul Erdős said

God may not play dice with the universe, but something strange is going on with the prime numbers. [Referring to Albert Einstein's famous belief that "God does not play dice with the universe."]

In a 1975 lecture, Don Zagier commented

There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision. (Havil 2003, p. 171)

Gaps between primes

Let p_n denote the nth prime number (i.e. p₁ = 2, p₂ = 3, etc.). The gap g_n between the consecutive primes p_n and p_{n + 1} is the difference between them, i.e.

g_n = p_{n + 1} − p_n.

We have g₁ = 3 − 2 = 1, g₂ = 5 − 3 = 2, g₃ = 7 − 5 = 2, g₄ = 11 − 7 = 4, and so on. The sequence (g_n) of prime gaps has been extensively studied.

For any natural number N larger than 1, the sequence (for the notation N! read factorial)

N! + 2, N! + 3, ..., N! + N

is a sequence of N − 1 consecutive composite integers. Therefore, there exist gaps between primes which are arbitrarily large, i.e. for any natural number N, there is an integer n with g_n > N. (Choose n so that p_n is the greatest prime number less than N! + 2.)

On the other hand, the gaps get arbitrarily small in proportion to the primes: the quotient g_n/p_n approaches zero as n approaches infinity. Note also that the twin prime conjecture asserts that g_n = 2 for infinitely many integers n.

Location of the largest known prime

The largest known prime, as of December 2005, is 2^30402457 − 1 (this number is 9,152,052 digits long); it is the 43rd known Mersenne prime. M_30402457 was found on December 15, 2005 by Curtis Cooper and Steven Boone, professors at Central Missouri State University and members of a collaborative effort known as GIMPS. Before finding the prime, Cooper and Boone ran the GIMPS program on a peak of 700 CMSU computers for 9 years.

The next two largest known primes are also Mersenne Primes: M_25964951=2^25964951 − 1 (42nd known Mersenne prime, 7,816,230 digits long) and M_24036583=2^24036583 − 1 (41st known Mersenne prime, 7,235,733 digits long). Historically, the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers, because there exists a particularly fast primality test for numbers of this form, the Lucas-Lehmer test for Mersenne numbers.

The largest known prime that is not a Mersenne prime is 27653 × 2^9167433 + 1 (2,759,677 digits). This is also the sixth largest known prime of any form. It was found by the Seventeen or Bust project and it brings them one step closer to solving the Sierpinski problem.

Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256^k for some positive integer k, and searching for possible primes within the interval [256^kn + 1, 256^k(n + 1) − 1].

Generalizations of the prime concept

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics.

Prime elements in rings

One can define prime elements and irreducible elements in any integral domain. For the ring Z of integers, the set of prime elements equals the set of irreducible elements; it's .

As an example, we consider the Gaussian integers Z[i], that is, complex numbers of the form a + bi with a and b in Z. This is an integral domain, and its prime elements are the Gaussian primes. Note that 2 is not a Gaussian prime, because it factors into the product of the two Gaussian primes (1 + i) and (1 − i). The element 3, however, remains prime in the Gaussian integers. In general, rational primes (i.e. prime elements in the ring Z of integers) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.

Prime ideals

In ring theory, one generally replaces the notion of number with that of ideal. Prime ideals are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ...

A central problem in algebraic number theory is how a prime ideal factors when it is lifted to an extension field. For example, in the Gaussian integer example above, (2) ramifies into a prime power (1 + i and 1 − i generate the same prime ideal), prime ideals of the form (4k + 3) are inert (remain prime), and prime ideals of the form (4k + 1) split (are the product of 2 distinct prime ideals).

Primes in valuation theory

In class field theory yet another generalization is used. Given an arbitrary field K, one considers valuations on K, certain functions from K to the real numbers R. Every such valuation yields a topology on K, and two valuations are called equivalent if they yield the same topology. A prime of K (sometimes called a place of K) is an equivalence class of valuations. With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic valuations on Q, for every prime number p.

Open questions

There are many open questions about prime numbers. The most significant of these is the Riemann hypothesis, which essentially says that the primes are as regularly distributed as possible. 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 1/ log x of number 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.

Other famous conjectures have a much greater chance of being true (in a formal sense, they follow from simple heuristic probabilistic arguments) with the lack of a solution more likely a reflection of the lack of suitable proof tools:

• Strong Goldbach conjecture: Every even integer greater than 2 can be written as a sum of two primes.

• Weak Goldbach conjecture: Every odd integer greater than 5 can be written as a sum of three primes.

• Twin prime conjecture: There are infinitely many twin primes, pairs of primes with difference 2, such as 11 and 13.

• Polignac's conjecture: For every integer n, there are infinitely many pairs of consecutive primes which differ by 2n. (When n=1 this is the twin prime conjecture.)

• Every even number is the difference of two primes.

• The Fibonacci sequence is believed by many to contain an infinite number of primes.

• There are infinitely many Mersenne primes but not Fermat primes.

• There are infinitely many primes of the form n² + 1.

• Legendre's conjecture: There is always a prime number between n² and (n + 1)² for every n.

• Cramer's conjecture: [ d(x)], the largest gap between consecutive primes, among all primes less than x, is asymptotic to [ \log^2 x]. This conjecture clearly implies the previous one, but its status is a little more unsure.

• Brocard's conjecture: There are always at least four primes between the squares of successive primes > 2.

Applications

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. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years. G. H. Hardy, A Mathematician's Apology 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.

Public-key cryptography

Several public-key cryptography algorithms, such as RSA, are based on extremely large prime numbers (that is, greater than 10¹⁰⁰).

Prime numbers in nature

Many numbers occur in nature, and inevitably some of these are prime. There are, however, relatively few examples of numbers that appear in nature because they are prime. For example, starfish have 5 'legs', which is a prime number. However there is no evidence to suggest that starfish have 5 'legs' because 5 is a prime number; insects seem to manage just fine with 6 legs.

One example of the use of prime numbers in nature is as an evolutionary strategy used by different species of insects called cicadas (both from the subspecies magicicada) which emerge in Eastern North America every 13 or 17 years to breed. The logic for this in nature appears to be that a predator specializing in magicicadas would have to emerge at the exact same frequency to find its prey.Paulo R. A. Campos, Viviane M. de Oliveira, Ronaldo Giro, and Douglas S. Galvão. [Emergence of Prime Numbers as the Result of Evolutionary Strategy]. Phys. Rev. Lett. 93, 098107 (2004) If magicicadas had appeared 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.The Economist. [Invasion of the Brood]. 6 May 2004. Though small, this appears enough to drive natural selection towards a prime-numbered life-cycle in this insect.

Primes in pop culture

In an episode of , two mutually unintelligible sentient life forms use the beginning sequence of prime numbers to communicate the fact that they are intelligent, thinking beings.

In Stephen King's novels The Waste Lands and Wizard and Glass, the protagonists solve a puzzle using primes in order to ride on the sentient and deranged train, Blaine the Mono.

In Mark Haddon's novel The Curious Incident of the Dog in the Night-time, the main character is fascinated by prime numbers, and in the book the chapters are numbered with primes.

In the Stargate Atlantis episode "Hot Zone", physicists Rodney McKay and Radek Zelenka play the game "Prime or Not Prime" in which players randomly name numbers and expect the other player or players to determine whether the number is prime.

In Carl Sagan's Contact a message from outer space is sent to humanity. This message is hashed into pieces and comes in intervals of incremented prime numbers from 2 to 101.

In the movie Cube, prime numbers are used by the characters to escape from a mysterious facility.

Trivia

As a publicity stunt against the Digital Millennium Copyright Act and other WIPO Copyright Treaty implementations, some people have tried to encode various forms of DeCSS code with prime numbers, creating the set of illegal prime numbers. Such numbers, when converted to binary and executed as a computer program, perform acts encumbered by applicable law in one or more jurisdictions.

See also

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• Divisor
• Highly composite number
• Irreducible polynomial
• Logarithmic integral function

| width="" align="" valign="" style="padding-left:;"|

• Prime power
• Primorial
• Sphenic number
• Ulam spiral

|}

Notes

References

• John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press; 448 pages
• T. Gowers, Mathematics: A Very Short Introduction. Oxford University Press, 2002.
• H. Riesel, Prime Numbers and Computer Methods for Factorization, 2nd ed., Birkhäuser 1994.
• Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. HarperCollins; 352 pages. ISBN 0066210704. [The Music of Primes website].
• Karl Sabbagh, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics. Farrar, Straus and Giroux; 340 pages

External links

• The prime pages — http://primes.utm.edu/
• , [Prime Number] at MathWorld.
  • [MacTutor history of prime numbers]
  • [Prime Number Generator] - Generate a given number of primes above a given start number.
  • [The first 15, 000, 000 primes]
  • [The prime puzzles]
  • [An English translation of Euclid's proof that there are infinitely many primes]
  • [Primes] from WIMS is an online prime generator.
  • [Number Spiral with prime patterns]
  • [An Introduction to Analytic Number Theory, by Ilan Vardi and Cyril Banderier]
  • [Factorizer] Windows software to find prime numbers and pairs of prime numbers less than 2,147,483,646.
  • [A fast, downloadable prime number calculator written in Visual Basic 6]

   

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