# Category: Conjectures about prime numbers

Artin's conjecture on primitive roots
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. The conjecture also as
Goldbach's weak conjecture
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expre
Bunyakovsky conjecture
The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial in one variable with integer coefficients to give infinitely many prime values in the sequence It was stated
Agrawal's conjecture
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: Let and be two coprime positive integers. If
Elliott–Halberstam conjecture
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 is named for Peter
Legendre's constant
Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function . Its value is now known to
Mersenne conjectures
In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one.
Oppermann's conjecture
Oppermann's conjecture is an unsolved problem in mathematics on the distribution of prime numbers. It is closely related to but stronger than Legendre's conjecture, Andrica's conjecture, and Brocard's
Waring–Goldbach problem
The Waring–Goldbach problem is a problem in additive number theory, concerning the representation of integers as sums of powers of prime numbers. It is named as a combination of Waring's problem on su
Dickson's conjecture
In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by Dickson that for a finite set of linear forms a1 + b1n, a2 + b2n, ..., ak + bkn with bi ≥ 1, there are infin
Fortune's conjecture
No description available.
Second Hardy–Littlewood conjecture
In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the first Hardy–Littlewood conjecture, the second Hardy–Littlewood conjecture was propos
Twin prime conjecture
No description available.
Waring's prime number conjecture
In number theory, Waring's prime number conjecture is a conjecture related to Vinogradov's theorem, named after the English mathematician Edward Waring. It states that every odd number exceeding 3 is
Andrica's conjecture
Andrica's conjecture (named after Dorin Andrica) is a conjecture regarding the gaps between prime numbers. The conjecture states that the inequality holds for all , where is the nth prime number. If d
Gilbreath's conjecture
Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and
Schinzel's hypothesis H
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 the twin prime conje
First Hardy–Littlewood conjecture
No description available.
Brocard's conjecture
In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn is the nth prime number, for every n ≥ 2. The conjecture is name
Agoh–Giuga conjecture
In number theory the Agoh–Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if It is named after and .
Dubner's conjecture
Dubner's conjecture is an as yet (2018) unsolved conjecture by American mathematician Harvey Dubner. It states that every even number greater than 4208 is the sum of two t-primes, where a t-prime is a
Bateman–Horn conjecture
In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger
Chinese hypothesis
In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that is divisible by n—in other words, that an integer n
Cramér's conjecture
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps betw
Grimm's conjecture
In mathematics, and in particular number theory, Grimm's conjecture (named after Carl Albert Grimm, 1 April 1926 – 2 January 2018) states that to each element of a set of consecutive composite numbers
Lemoine's conjecture
In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an od
Firoozbakht's conjecture
In number theory, Firoozbakht’s conjecture (or the Firoozbakht conjecture) is a conjecture about the distribution of prime numbers. It is named after the Iranian mathematician Farideh Firoozbakht from
Landau's problems
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 at the present stat
Polignac's conjecture
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 words: There are i
Pólya conjecture
In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors. The conject
Legendre's conjecture
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between and for every positive integer . The conjecture is one of Landau's problems (1912) on prime number
Goldbach's conjecture
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 sum of two prime num