Abhyankar's conjecture | Baudet's conjecture | bellows conjecture | Bieberbach conjecture | Calabi conjecture | Catalan's conjecture | Kepler conjecture | Mordell conjecture | Poincaré conjecture | Taniyama-Shimura conjecture | Thurston's geometrization conjecture (total: 11)

For a finite group G, let p(G) be the subgroup generated by all the Sylow p-subgroups of G. Then Abhyankar's conjecture posits that if X is a projective curve in characteristic p>0, and if x_0, x_1, ..., x_t are points of X (for t>0), then a necessary and sufficient condition that G occur as the Galois group of a finite covering Y of X, branched only at the points x_0, x_1, ..., x_t, is that the quotient group G/(p(G)) has 2g + t generators.

Baudet's conjecture posits that if C_1, C_2, ..., C_r are sets of positive integers and union _(i = 1)^rC_i = Z^+, then some C_i contains arbitrarily long arithmetic progressions.

The bellows conjecture posits that all flexible polyhedra keep a constant volume as they are flexed.

The Bieberbach conjecture, now proved, posited that the nth coefficient in the power series of a univalent function is no greater than n.

The Calabi conjecture states that a compact Kähler manifold has a unique Kähler metric in the same class whose Ricci form is any given 2-form representing the first Chern class. In particular, if the first Chern class vanishes, then there is a unique Kähler metric in the same class with vanishing Ricci curvature.

Catalan's conjecture states that 8 and 9, (2^3 and 3^2) are the only consecutive powers (excluding 0 and 1).

The Kepler conjecture, now proved, posited that cubic and hexagonal close packing are the densest possible sphere packings.

The Mordell conjecture, now proved, posited that Diophantine equations that give rise to surfaces with two or more holes have only finitely many solutions in Gaussian integers with no common factors.

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

The Taniyama-Shimura conjecture, now proved, posited that there exists a modular form of weight two and level N which is an eigenform under the Hecke operators and has a Fourier series sum_(n=1)^∞a_nq^n.

Thurston's geometrization conjecture, now proved, posited that after a three-manifold is split into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of eight specific geometries.

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

| formal statement Catalan's conjecture | ¬exists_({p, q, m, n}, (p, q, m, n) element Z^4 ∧ p>1 ∧ q>1 ∧ m>1 ∧ n>1 ∧ q^n!=8)p^m - q^n = 1

| formulation date | formulators | status Abhyankar's conjecture | 1957 (68 years ago) | S. S. Abhyankar | proved Baudet's conjecture | 1926 (99 years ago) | PJH Baudet | proved bellows conjecture | | Dennis Sullivan | proved Bieberbach conjecture | 1916 (109 years ago) | Ludwig Georg Elias Moses Bieberbach | proved Calabi conjecture | 1954 (71 years ago) | Eugenio Calabi | proved Catalan's conjecture | 1844 (181 years ago) | Eugène Charles Catalan | proved Kepler conjecture | 1611 (414 years ago) | Johannes Kepler | proved Mordell conjecture | 1922 (103 years ago) | Louis Joel Mordell | proved Poincaré conjecture | 1904 (121 years ago) | Henri Poincaré | proved Taniyama-Shimura conjecture | 1955 (70 years ago) | Yutaka Taniyama | Goro Shimura | proved Thurston's geometrization conjecture | 1982 (43 years ago) | William Paul Thurston | proved | proof date | provers Abhyankar's conjecture | 1994 (37 years later) (31 years ago) | David Harbater Baudet's conjecture | 1928 (2 years later) (97 years ago) | Bartel Leendert van der Waerden bellows conjecture | 1997 (28 years ago) | Bob Connelly | Idzhad Sabitov | Anke Walz Bieberbach conjecture | 1985 (69 years later) (40 years ago) | Louis de Branges Calabi conjecture | 1976 (22 years later) (49 years ago) | Shing-Tung Yau Catalan's conjecture | 2002 (158 years later) (23 years ago) | Preda Mihăilescu Kepler conjecture | 2005 (394 years later) (20 years ago) | Thomas Callister Hales Mordell conjecture | 1983 (61 years later) (42 years ago) | Gerd Faltings Poincaré conjecture | 2003 (99 years later) (22 years ago) | Grigori Perelman Taniyama-Shimura conjecture | 2001 (46 years later) (24 years ago) | Christophe Breuil | Brian Conrad | Fred Diamond | Richard Taylor Thurston's geometrization conjecture | 2003 (21 years later) (22 years ago) | Grigori Perelman | additional people involved Taniyama-Shimura conjecture | André Weil

p^m - q^n!=1

| prizes offered for solution | prizes awarded for solution Mordell conjecture | | Faltings shared the 1986 Fields Medal for his proof of the Mordell conjecture. Poincaré conjecture | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. (Refused by solver.) | Perelman shared the 2006 Fields Medal for his proof. (Refused by solver.) Thurston's geometrization conjecture | | Perelman shared the 2006 Fields Medal for his proof of the Poincaré conjecture. (Refused by solver.)

proved conjectures | solved mathematics problems | mathematics theorems