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 + Bm = Cl 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 xuyv = 1 in terms of integers; Formulated in 1844, Mihailescu has shown that all integer solutions to xuyv = 1 must have u and v as Double Weiferich primes, that is, pq–1 = 1 (mod p2) and qp–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 abc(00)apr].
@ References: Wang 02; Sanchis-Lozano et al IJMPA(12)-a1202 [and quantum field theory]; Castelvecchi SA(12)may [nearing solution].
> Online resources: see MathWorld page; Wikipedia page.

Langlands Conjecture
* Idea: A relationship between perfect squares and modular arithmetic conjectured by R Langlands in the 1960s; 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]; 2010, Clay Mathematics Institute Millennium Prize awarded to Grigory Perelman, who turned it down [@ news pt(10)jul].
@ 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: see Wikipedia page.

Riemann Hypothesis / Conjecture > s.a. Zeta Function.
* Idea: A conjecture on how prime numbers are distributed amongst other numbers; All of the non-trivial zeros of the Riemann zeta function ζ(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.
* And physics / spectral approach: An approach has been developed whose goal is to lead to a physicist's proof of the Riemann hypothesis by providing a realization of the non-trivial zeros of the Riemann zeta function as the spectrum of the Hamiltonian of a massless Dirac fermion in a region of Rindler spacetime containing moving mirrors whose accelerations are related to the prime numbers.
@ General references: Sabbagh 04 [r pw(03)apr]; Fujimoto & Uehara a0906, a1003; Castro IJGMP(10) [two approaches]; McPhedran a1309; Mazur & Stein 16 [and prime numbers].
@ And physics: Acharya a0903 [quantum-mechanical model]; Sierra a1012-in [spectral approach]; Planat et al JPA(11)-a1012; Schumayer & Hutchinson RMP(11)-a1101; Srednicki PRL(11)-a1105; Vericat PhyA(13)-a1211 [and the classical statistical mechanics of a lattice gas]; Wolf a1410; Sierra a1601/JPA [rev]; > s.a. number theory [prime numbers].
@ 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.
@ General references: Hisano & Sornette MI(13)-a1202 [on the distribution of time-to-proof's for mathematical conjectures].
> Other: see graph theory [Gallai conjecture, Wagner conjecture]; Gromov-Lawson-Rosenberg Conjecture; mathematics [Shimura-Taniyama-Weil conjecture]; Schreier Conjecture; Smale Conjecture; sphere [annulus, Kepler, Smith conjectures]; vector fields [Weinstein conjecture].

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