Wilson's theorem

Wilson's theorem - First proof. This proof uses the fact that if p is an odd prime, then the set of numbers G = (Z/pZ)× = {1, 2, ... p − 1} forms a group under multiplication modulo p. This means that for each element i in G, there is a unique inverse element j in G such that ij ≡ 1 (mod p). If i ≡ j (mod p), then i2 ≡ 1 (mod p), which forces i2See also:

Wilson's theorem, Wilson's theorem - History, Wilson's theorem - Proofs, Wilson's theorem - First proof, Wilson's theorem - Second proof, Wilson's theorem - Applications, Wilson's theorem - Generalization, Wilson's theorem - Converse

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A composite number is a positive integer which has a positive divisor other than one or itself. By definition, every integer greater than one is either a prime number or a composite number. The numbers zero and one are considered to be neither prime nor composite. The integer 14 is a composite number because it can be factored as 2 Ã— 7. Composite number - Properties. All even numbers greater than 2 are composite numbers. The smallest composite number is 4. Every co ...

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• Composite number - Properties
• Composite number - Kinds of composite numbers

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In mathematics, a Wilson prime is a certain kind of prime number. A prime p is called a Wilson prime if p² divides (p − 1)! + 1, where ! denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1. The only known Wilson primes are 5, 13, and 563 (sequence A007540 in OEIS); if any others exist, they must be greater than 5 · 108. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [x, ...

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One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors. For the latter (where μ is the Möbius function and x is half the tot ...

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Composite number, Composite number - Properties, Composite number - Kinds of composite numbers

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A set of Diophantine equations in 26 variables can be used to obtain primes: A given number k + 2 is prime iff. the following system of Diophantine equations has a solution in the natural numbers (Riesel, 1994): 0 = wz + h + j − q 0 = (gk + 2g + k + 1)(h + j) + h − z 0 = (16k + 1)3(k + 2)(n ...

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Formula for primes, Formula for primes - Prime formulas and polynomial functions, Formula for primes - Formula based on a system of Diophantine equations, Formula for primes - Formulas using the floor function, Formula for primes - Mills's formula, Formula for primes - Floor function formulas based on Wilson's theorem, Formula for primes - Another approach

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Joseph Louis Lagrange - Early years. He was born (as Giuseppe Luigi Lagrangia) in Turin. His father, who had charge of the Kingdom of Sardinia's military chest, was of good social position and wealthy, but before his son grew up he had lost most of his property in speculations, and young Lagrange had to rely on his own abilities for his position. He was educated at the college of Turin, but it was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excit ...

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Joseph Louis Lagrange, Joseph Louis Lagrange - Biography, Joseph Louis Lagrange - Early years, Joseph Louis Lagrange - Middle years, Joseph Louis Lagrange - Later years, Joseph Louis Lagrange - Appearance, Joseph Louis Lagrange - Pure mathematics

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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 (The proof of this is below). For example, we can write and any other such factorization of 23244 will be identical except for the order of the factors. See prime fac ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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List of number theory topics - Primality tests. Prime factorization algorithm Trial division Sieve of Eratosthenes Probabilistic algorithm Fermat primality test Pseudoprime Carmichael number Euler pseudoprime Euler-Jacobi pseudoprime Fibonacci pseudoprime Probable prime Miller-Rabin primality test Lucas-Lehmer primality test Lucas-Lehmer test for Mersenne numbers AKS prim ...

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List of number theory topics, List of number theory topics - Factors, List of number theory topics - Fractions, List of number theory topics - Modular arithmetic, List of number theory topics - Arithmetic functions, List of number theory topics - Analytic number theory: additive problems, List of number theory topics - Algebraic number theory, List of number theory topics - Quadratic forms, List of number theory topics - L-functions, List of number theory topics - Diophantine equations, List of number theory topics - Diophantine approximation, List of number theory topics - Sieve methods, List of number theory topics - Named primes, List of number theory topics - Combinatorial number theory, List of number theory topics - Computational number theory, List of number theory topics - Primality tests, List of number theory topics - Integer factorization, List of number theory topics - Pseudo-random numbers, List of number theory topics - History

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It is known that no non-constant polynomial function P(n) exists that evaluates to a prime number for all integers n. The proof is simple: Suppose such a polynomial existed. Then P(1) would evaluate to a prime p, so . But for any k, also, so P(1 + kp) cannot also be prime (as it would be divisible by p) unless it were p itself, but the only way P(1 + kp) = ...

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Formula for primes, Formula for primes - Prime formulas and polynomial functions, Formula for primes - Formula based on a system of Diophantine equations, Formula for primes - Formulas using the floor function, Formula for primes - Mills's formula, Formula for primes - Floor function formulas based on Wilson's theorem, Formula for primes - Another approach

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The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Prime number - 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 {...−11, −7, −5, −3, −2, 2, 3, 5, 7, 11, ...}. As an example, we consider the Gaussian integers Z[i], that is, complex numbers o ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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Main article formula for primes There is no formula for primes which is more efficient at finding primes than the methods mentioned above under "Finding prime numbers". Those which do exist have little practical value. The curious polynomial f(n) = n2 âˆ’ n + 41 yields primes for n = 0,..., 40, but f(41) is composite. It has been proved that there i ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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Let pn denote the n-th prime number (i.e. p1 = 2, p2 = 3, etc.). The gap gn between the consecutive primes pn and pn + 1 is the number of (composite) numbers between them, i.e. gn = pn + 1 ∠...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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There are infinitely many prime numbers. The oldest known proof for this statement 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 o ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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Lagrange's interests were essentially those of a student of pure mathematics: he sought and obtained far-reaching abstract results, and was content to leave the applications to others. Indeed, no inconsiderable part of the discoveries of his great contemporary, Laplace, consists of the application of the Lagrangian formulae to the facts of nature; for example, Laplace's conclusions on the velocity of sound and the secular acceleration of the Moon are implicitly involved in Lagrange's results. The only difficulty in understanding Lagrange is ...

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Joseph Louis Lagrange, Joseph Louis Lagrange - Biography, Joseph Louis Lagrange - Early years, Joseph Louis Lagrange - Middle years, Joseph Louis Lagrange - Later years, Joseph Louis Lagrange - Appearance, Joseph Louis Lagrange - Pure mathematics

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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! âˆ’ 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,... (sequence A002982 in OEIS) n! + 1 is prime for n = 1, 2, 3, 11, 27, ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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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 if 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 so ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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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 a ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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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 Lucas-Lehmer primality 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 ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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The largest known prime, as of December 2005, is 230402457 âˆ’ 1 (this number is 9,152,052 digits long); it is the 43rd known Mersenne prime. M30402457 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. The next largest known prime is 225964951 âˆ’ 1 (this number is 7,816,230 digits long); it is the 42nd known Mersenne prime. M25964951 was found on Feb ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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There is also a generalization of Wilson's theorem, due to Carl Friedrich Gauss: where p is an odd prime. ...

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Wilson's theorem, Wilson's theorem - History, Wilson's theorem - Proofs, Wilson's theorem - First proof, Wilson's theorem - Second proof, Wilson's theorem - Applications, Wilson's theorem - Generalization, Wilson's theorem - Converse

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