abc conjecture | Andrica's conjecture | Barnette's conjecture | Beal's conjecture | de Polignac's conjecture | Eberhart's conjecture | Erdős-Turán conjecture | Fuglede's conjecture | strong Goldbach conjecture | Hodge conjecture | Jacobian conjecture | Keller's conjecture | Legendre's conjecture | Quillen-Lichtenbaum conjecture | Scholz conjecture | Sierpiński's conjecture | Swinnerton-Dyer conjecture | twin prime conjecture | second twin prime conjecture | strong twin prime conjecture | Wagstaff's conjecture (total: 21)

For any ϵ>0, there exists a constant C_ϵ such that for any three relatively prime integers a, b, c satisfying a + b = c, the inequality max( left bracketing bar a right bracketing bar , left bracketing bar b right bracketing bar , left bracketing bar c right bracketing bar )<=C_ϵ product_(p|abc)p^(1 + ϵ) holds.

Andrica's conjecture posits that for p_n the nth prime number, sqrt(p_(n + 1)) - sqrt(p_n)<1.

Barnette's conjecture posits that every 3-connected bipartite cubic planar graph is Hamiltonian.

Beal's conjecture posits that if a^x + b^y = c^z, where a, b, c, x, y, and z are any positive integers with x, y, z>2, then a, b, and c have a common factor.

de Polignac's conjecture posits that every even number is the difference of two consecutive primes in infinitely many ways.

Defining q_n as the nth prime such that M_(q_n) is a Mersenne prime, then Eberhart's conjecture posits that q_n~(3/2)^n.

The Erdős-Turán conjecture states that if the sum of reciprocals of a set of integers diverges, then that set contains arbitrarily long arithmetic progressions.

Fuglede's conjecture posits that a domain Ω admits an operator spectrum iff it is possible to tile R^d by a family of translates of Ω.

The strong Goldbach conjecture posits that every even integer greater than two is the sum of two primes.

The Hodge conjecture posits that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually rational linear combinations of geometric pieces called algebraic cycles.

| formal statement abc conjecture | for all _(ϵ, ϵ>0)exists_(C_ϵ) for all _({a, b, c}, (a, b, c) element Z^3 ∧ gcd(a, b, c) = 1 ∧ a + b = c)max( left bracketing bar a right bracketing bar , left bracketing bar b right bracketing bar , left bracketing bar c right bracketing bar )<=C_ϵexp( sum_(n∣abc ∧ n element P)(1 + ϵ)log(n)) Andrica's conjecture | for all _(n, n element Z^+)sqrt(p_(n + 1)) - sqrt(p_n)<1 Beal's conjecture | for all _({a, b, c, x, y, z}, a^x + b^y = c^z ∧ (a, b, c, x, y, z) element (Z^+)^6 ∧ (x|y|z)>2)gcd(a, b, c)>1 de Polignac's conjecture | for all _(k, k element Z^+) sum_(n=1)^∞Boole(p_(n + 1) - p_n = 2k) = ∞ strong Goldbach conjecture | for all _(n, n element Z ∧ n>2)exists_({i, j}, (i, j) element (Z^+)^2)2n = p_i + p_j Legendre's conjecture | for all _(n, n element Z^+)exists_(k, k element Z^+)n^2

| formulation date | formulators | status abc conjecture | 1985 (40 years ago) | Joseph Oesterlé | David Masser | open Andrica's conjecture | 1986 (39 years ago) | Dorin Andrica | open Barnette's conjecture | 1969 (56 years ago) | David Barnette | open Beal's conjecture | 1997 (28 years ago) | Andrew Beal | open de Polignac's conjecture | 1849 (176 years ago) | Alphonse de Polignac | open Eberhart's conjecture | | Eberhart | open Erdős-Turán conjecture | 1936 (89 years ago) | Paul Erdős | Paul Turán | open Fuglede's conjecture | 1974 (51 years ago) | Bent Fuglede | open strong Goldbach conjecture | | Leonhard Euler | open Hodge conjecture | 1950 (75 years ago) | William Vallance Douglas Hodge | open Jacobian conjecture | 1939 (86 years ago) | Ott-Heinrich Keller | open Keller's conjecture | 1930 (95 years ago) | Ott-Heinrich Keller | refuted Legendre's conjecture | | Adrien-Marie Legendre | open Quillen-Lichtenbaum conjecture | | | open Scholz conjecture | 1937 (88 years ago) | Arnold Scholz | open Sierpiński's conjecture | 1950 (75 years ago) | Wacław Sierpiński | proved Swinnerton-Dyer conjecture | 1960 (65 years ago) | Peter Swinnerton-Dyer | open twin prime conjecture | 1849 (176 years ago) | Alphonse de Polignac | open second twin prime conjecture | | | open strong twin prime conjecture | | | open Wagstaff's conjecture | 1983 (42 years ago) | Samuel S. Wagstaff | open | proof date | provers Keller's conjecture | 1992 (62 years later) (33 years ago) | Jeffrey Lagarias | Peter Shor | John Mackey | Jennifer Debroni Sierpiński's conjecture | 1998 (48 years later) (27 years ago) | Kevin B. Ford | additional people involved Keller's conjecture | Oskar Perron