Beal's conjecture | Collatz problem | Erdős-Turán conjecture | Fermat's last theorem | strong Goldbach conjecture | Hodge conjecture | smooth solution to the Navier-Stokes equations problem | Poincaré conjecture | P vs. NP problem | Riemann hypothesis | Swinnerton-Dyer conjecture | Wolfram 2, 3 Turing machine research prize | Yang-Mills existence and mass gap problem (total: 13)

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.

The Collatz problem is determination of if iterating the recurrence a_n = piecewise | a_(n - 1)/2 | a_(n - 1) congruent 0 (mod 2) 3a_(n - 1) + 1 | a_(n - 1) congruent 1 (mod 2) always returns to 1 for positive a_0.

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.

Fermat's last theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

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.

The smooth solution to the Navier-Stokes equations problem asks if the Navier-Stokes equations on a 3-dimensional domain Ω in R^3 have a unique smooth solution for all time.

The Poincaré conjecture, now proved, posited that every simply connected closed three-manifold is homeomorphic to the three-sphere.

The P vs. NP problem asks for the determination of whether all NP-problems are actually P-problems.

The Riemann hypothesis posits that the nontrivial zeros of the Riemann zeta function ζ(s) all lie on the critical line Re(s) = 1/2.

| solution Wolfram 2, 3 Turing machine research prize | Yes.

The three-sphere is the only type of bounded three-dimensional space possible that contains no holes.

| formal statement 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 Fermat's last theorem | ¬exists_({a, b, c, n}, (a, b, c, n) element (Z^+)^4 ∧ n>2 ∧ abc!=0)a^n + b^n = c^n strong Goldbach conjecture | for all _(n, n element Z ∧ n>2)exists_({i, j}, (i, j) element (Z^+)^2)2n = p_i + p_j Riemann hypothesis | for all _(n, n element Z ∧ n!=0)Re(ρ_n) = 1/2

| formulation date | formulators | status Beal's conjecture | 1997 (29 years ago) | Andrew Beal | open Collatz problem | 1937 (89 years ago) | Lothar Collatz | open Erdős-Turán conjecture | 1936 (90 years ago) | Paul Erdős | Paul Turán | open Fermat's last theorem | 1637 (389 years ago) | Pierre de Fermat | proved strong Goldbach conjecture | | Leonhard Euler | open Hodge conjecture | 1950 (76 years ago) | William Vallance Douglas Hodge | open smooth solution to the Navier-Stokes equations problem | | | open Poincaré conjecture | 1904 (122 years ago) | Henri Poincaré | proved P vs. NP problem | | | open Riemann hypothesis | 1859 (167 years ago) | Bernhard Riemann | open Swinnerton-Dyer conjecture | 1960 (66 years ago) | Peter Swinnerton-Dyer | open Wolfram 2, 3 Turing machine research prize | 2002 (24 years ago) | Stephen Wolfram | proved Yang-Mills existence and mass gap problem | 1954 (72 years ago) | Chen Ning Yang | Robert L. Mills | open | proof date | provers Fermat's last theorem | 1995 (358 years later) (31 years ago) | Andrew Wiles Poincaré conjecture | 2003 (99 years later) (23 years ago) | Grigori Perelman Wolfram 2, 3 Turing machine research prize | Wednesday, October 24, 2007 | Alex Ian Smith | additional people involved Fermat's last theorem | Richard Taylor

a^x + b^y = c^z

a^n + b^n = c^n

2n = p_i + p_j

(du)/(dt) + u · del u = -( del P)/ρ + ν del ^2u

Re(ρ_n) = 1/2

Known to always reach 1 for n <= 19×2^58 ≈ 5.48×10^18.

Verified for all n < 4×10^18.

It has been verified that the first 1×10^13 nontrivial zeros of the zeta function lie on the critical line. Conrey (1989) proved that at least 40% of the nontrivial zeros of the zeta function lie on the critical line.