Mathematical Conjectures

Adams Conjecture
* Idea: An algebraic topology conjecture, proven by Quillen & Sullivan using étale cohomology.

Beal Conjecture (After a banker who offered a reward)
* Idea: A generalization of Fermat's Last Theorem.
$ Def: There are no solutions to the equation Aⁿ + B^m = C^l for A, B, C relative primes, and n, m, l 3.

Bieberbach Conjecture
* History: Proved by Louis de Branges ( 1977).

Catalan's Conjecture
* Idea and status: The numbers 8 and 9 are the only two consecutive powers of integers, or 3² – 2³ = 1 is the only solution to x^u – y^v = 1 in terms of integers; Formulated in 1844, Mihailescu has shown that all integer solutions to x^u – y^v = 1 must have u and v as Double Weiferich primes, that is, p^q–1 = 1 (mod p²) and q^p–1 = 1 (mod q²); 2000, A distributed computing effort is under way; 2004, Proved by the Swiss mathematician Preda Mihailescu.
@ References: Metsänkylä BAMS(04) [history].

Goldbach Conjecture
$ Def: Every even number is the sum of two primes.
* History: Proposed in 1742 by Prussian mathematician Christian Goldbach; 1996, Checked up to 2 × 10⁶; The U.S. publisher of Uncle Petros and Goldbach's Conjecture has promised $1 million to the first person to prove it, provided the proof appears in a reputable mathematics journal before 2004 [@ Paulos abcnews.com 04/00].

Langlands Conjecture
* Idea: A relationship between perfect squares and modular arithmetic conjectured by R Langlands in the 1960's; proved in 2000 [@ NAMS].

Poincaré's Conjecture > s.a. 3D manifolds.
* Idea: The only topological n-manifold with the same fundamental group and homology as Sⁿ is Sⁿ itself.
* History: The cases n = 1, 2, > 5 were proved long ago [@ Smale AJM(62)]; 1982, Proof for n = 4 by M Freedman [@ JDG(82)]; 2005, The case n = 3 (the one Poincaré originally proposed in 1904, with the remark "this question would lead us too far astray") is still unsolved, despite the 1986 claim by E Rego & C Rourke, but...; 2006, Fields medal awarded to Grigory Perelman, who presented a proof in 2002 that seems to be correct – Perelman did not show up for the awards ceremony in Madrid, effectively declining the prize [@ news BBC(06)aug].
@ References: Stewart Nat(86)mar, Nat(87)feb; Collins SA(04)jul [proved?]; Morgan BAMS(05) [progress]; news BBC(06)dec [proof]; Kholodenko JGP(08)ht/07 [towards physically motivated proofs]; O'Shea 07 [history, proof; r pw(07)aug Hitchin].
> Online resources: Wikipedia page.

Riemann Hypothesis / Conjecture
* Idea: A conjecture on how prime numbers are distributed amongst other numbers; All of the nontrivial zeros of the Riemann zeta function (s) are on the critical line Re(s) = 1/2.
* History: 1859, Published by Riemann; 2001, The Clay Mathematics Institute in Cambridge, MA, offered a $1M prize to whoever proves it first; 2004, Louis de Branges claims to have a proof.
@ References: Sabbagh 02 [r pw(03)apr].
@ Related topics: Okubo JPA(98) [and 2D Lorentz-invariant Hamiltonian]; Castro & Mahecha CSF(02)ht/00 [and fractal spacetime]; Derbyshire 03; Elizalde et al IJMPA(03)mp/01 [on strategies]; Bunimovich & Dettmann PRL(05) [and open circular billiards]; Coffey MPAG(05)mp, mp/05 [Li criterion, constants].

Weil Conjecture (Arithmetic)
* History: Proved by Deligne using étale cohomology.
@ References: Deligne IHES(74).

Other Conjectures and ex-Conjectures > s.a. Fermat's Last Theorem.
* Mordell conjecture: Proved by G Faltings.
* Robbins conjecture: Proved in 1996 by Woos & McCune by computer.
> Other: see Gallai; Gromov-Lawson-Rosenberg; math [Shimura-Taniyama-Weil]; Schreier; Smale; sphere [annulus, Kepler, Smith].

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