391 problems

In number theory, Skewes's number is the smallest natural number $x$ for which the prime-counting function $\pi (x)$ exceeds the logarithmic integral function…

In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined…

In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.

In mathematics, Wolstenholme's theorem states that for a prime number p ≥ 5, the congruence

In number theory, a Sierpiński number is an odd natural number k such that $k\times 2^{n}+1$ is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there ar…

In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalization of widely open conjectures such as…

In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states:For any positive even number n, there are infinitely many prime gaps of size n. In other…

In mathematics, the Mersenne conjectures concern the characterization of a kind of prime numbers called Mersenne primes, meaning prime numbers that are a power of two minus one.

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,…

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (17, 19) or (41, 43). In other words, a t…

Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between $n^{2}$ and $(n+1)^{2}$ for every positive integer $n$. The conjecture is one…

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the…

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable…

In number theory, Gillies' conjecture is a conjecture about the distribution of prime factors of Mersenne numbers. It was made by Donald B. Gillies in a 1964 paper in which he also…

In number theory, the Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive…

In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by Walter Feit and John G. Thompson (1962). The conjecture states that there are no distinc…

A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, t…

In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It…

In number theory, Dickson's conjecture is the statement that for a finite set of linear forms $a_{1}+b_{1}n,a_{2}+b_{2}n,\dots,a_{k}+b_{k}n$ with each ⁠ $b_{i}\geq 1$ ⁠, there are…

In mathematics, a double Mersenne number is a Mersenne number of the form $M_{M_{p}}=2^{2^{p}-1}-1$ where $p$ is prime.

The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial $f(x)$ in one variable with integer coefficients to give infinitely many prime values in…

In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. Th…

In number theory, Agrawal's conjecture, due to Manindra Agrawal in 2002, forms the basis for the cyclotomic AKS test. Agrawal's conjecture states formally:

In number theory, the Agoh–Giuga conjecture on the Bernoulli numbers $B_{k}$ postulates that $p$ is a prime number if and only if $pB_{p-1}\equiv -1\!{\pmod {p}}.$ It is named afte…

A natural number is called even if it is divisible by $2$, and a prime is a natural number greater than $1$ with no positive divisors other than $1$ and itself. Goldbach's conjectu…