Borsuk's conjecture | Feit-Thompson conjecture | Keller's conjecture | Mertens conjecture | Pólya conjecture | Seifert conjecture | Tait's conjecture | Wang's conjecture (total: 8)

Borsuk's conjecture posits that it is possible to cut an n-dimensional shape of generalized diameter 1 into n + 1 pieces, each with diameter smaller than the original.

The Feit-Thompson conjecture (now refuted) posited that there are no primes p and q for which (p^q - 1)/(p - 1) and (q^p - 1)/(q - 1) have a common factor.

Keller's conjecture, now refuted, posited that tiling an n-dimensional space with n-dimensional hypercubes of equal size always yields an arrangement in which at least two hypercubes have an entire (n - 1)-dimensional "side" in common.

The Mertens conjecture, now refuted, posited that for all integers n>1, left bracketing bar M(n) right bracketing bar

The Pólya conjecture, now refuted, posited that the summatory function L(m) = sum_(n=1)^mλ(n) of the Liouville function is always nonpositive.

The Seifert conjecture posits that every smooth nonzero vector field on the 3-sphere has at least one closed orbit.

Tait's conjecture, now refuted, posited that every cubic polyhedral graph is Hamiltonian.

Wang's conjecture, now refuted, posited that if a set of tiles can tile the plane, then they can always be arranged to do so periodically.

| solution Borsuk's conjecture | False Feit-Thompson conjecture | False Mertens conjecture | False Pólya conjecture | False Seifert conjecture | False Tait's conjecture | False Wang's conjecture | False

| formal statement Feit-Thompson conjecture | ¬exists_({p, q}, (p, q) element P^2)gcd((q^p - 1)/(p - 1), (q^p - 1)/(q - 1))>1 Mertens conjecture | for all _(n, n element Z ∧ n>1) left bracketing bar sum_(k=1)^nμ(k) right bracketing bar

| formulation date | formulators | status Borsuk's conjecture | 1932 (94 years ago) | Karol Borsuk | refuted Feit-Thompson conjecture | 1962 (64 years ago) | Walter Feit | John Griggs Thompson | refuted Keller's conjecture | 1930 (96 years ago) | Ott-Heinrich Keller | refuted Mertens conjecture | 1885 (141 years ago) | Thomas Jan Stieltjes | refuted Pólya conjecture | 1919 (107 years ago) | George Pólya | refuted Seifert conjecture | 1950 (76 years ago) | Herbert Seifert | refuted Tait's conjecture | 1880 (146 years ago) | Peter Tait | refuted Wang's conjecture | 1961 (65 years ago) | Hao Wang | refuted | proof date | provers Borsuk's conjecture | 1993 (61 years later) (33 years ago) | Jeff Kahn | Gil Kalai Feit-Thompson conjecture | 1971 (9 years later) (55 years ago) | Nelson M Stephens Keller's conjecture | 1992 (62 years later) (34 years ago) | Jeffrey Lagarias | Peter Shor | John Mackey | Jennifer Debroni Mertens conjecture | 1985 (100 years later) (41 years ago) | Andrew Odlyzko | Herman J.J. te Riele Pólya conjecture | 1958 (39 years later) (68 years ago) | C. Brian Haselgrove Seifert conjecture | 1994 (44 years later) (32 years ago) | Krystyna M. Kuperberg Tait's conjecture | 1946 (66 years later) (80 years ago) | William Thomas Tutte Wang's conjecture | 1966 (5 years later) (60 years ago) | Robert Berger | additional people involved Keller's conjecture | Oskar Perron

gcd((q^p - 1)/(p - 1), (q^p - 1)/(q - 1))>1

left bracketing bar sum_(k=1)^nμ(k) right bracketing bar

sum_(n=1)^mλ(n)<=0

Perron (1940) proved Keller's conjecture to be true in dimensions six and less. The conjecture has been shown to be false in dimensions 8, 10, and 12 by Lagarias and Shor (1992), who found a cliques of size and in the 10- and 12-dimensional Keller graphs, respectively, and by Mackey (2002), who found a clique of size in the Keller graph of dimension eight. Debroni et al. (2011) recently showed the clique number of the 7-dimensional Keller graph is 124, thus suggesting (but not establishing) that Keller's conjecture is false in that dimension.

The value of the smallest counterexample is still unknown but known to exceed 1×10^14.

The smallest counterexample is known to occur at m = 906150257.

refuted conjectures