# Category: Theorems about prime numbers

Green–Tao theorem
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for e
Brun's theorem
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually deno
Brun–Titchmarsh theorem
In analytic number theory, the Brun–Titchmarsh theorem, named after Viggo Brun and Edward Charles Titchmarsh, is an upper bound on the distribution of prime numbers in arithmetic progression.
Divergence of the sum of the reciprocals of the primes
The sum of the reciprocals of all prime numbers diverges; that is: This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime number
Maier's theorem
In number theory, Maier's theorem is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer. The theorem states that if π is the
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, the number is an integer multiple of p. In the notation of modular arithmetic, this is expressed as For example, if
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, w
Friedlander–Iwaniec theorem
In analytic number theory the Friedlander–Iwaniec theorem states that there are infinitely many prime numbers of the form . The first few such primes are 2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241,
Chen's theorem
In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes).
Proth's theorem
In number theory, Proth's theorem is a primality test for Proth numbers. It states that if p is a Proth number, of the form k2n + 1 with k odd and k < 2n, and if there exists an integer a for which th
Euler's criterion
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then Euler
Bonse's inequality
In number theory, Bonse's inequality, named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that if p1, ..., pn, pn+1 a
Mertens' theorems
In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens. "Mertens' theorem" may also refer to his theorem in analysis.
Rosser's theorem
In number theory, Rosser's theorem states that the nth prime number is greater than . It was published by J. Barkley Rosser in 1939. Its full statement is: Let pn be the nth prime number. Then for n ≥
In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's weak conjectu
Lucas's theorem
In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first a
Vantieghems theorem
In number theory, Vantieghems theorem is a primality criterion. It states that a natural number n(n≥3) is prime if and only if Similarly, n is prime, if and only if the following congruence for polyno
Bertrand's postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer , there always exists at least one prime number with A less restrictive formulation is: for every , there is always at
Fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquel
Lagrange's theorem (number theory)
In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, i
Wolstenholme's theorem
In mathematics, Wolstenholme's theorem states that for a prime number , the congruence holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one m
Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n
Erdős–Kac theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω(n) is the number of distinct pr
Euclid's lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: Euclid's lemma — If a prime p divides the product ab of two integers a and b, the
Prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common a
Proof of Bertrand's postulate
In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Chebyshev, and a short but advanced proof was given by Ramanujan. The
Linnik's theorem
Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive c and L such that, if we denote p(a,d)
Siegel–Walfisz theorem
In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both
Euclid's theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work Elements. There are several proofs of
Hardy–Ramanujan theorem
In mathematics, the Hardy–Ramanujan theorem, proved by G. H. Hardy and Srinivasa Ramanujan, states that the normal order of the number ω(n) of distinct prime factors of a number n is log(log(n)). Roug