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
*A*^{n}
+ *B*^{m}
= *C*^{l} for *A*, *B*,
*C* relative primes, and *n*, *m*, *l* ≥ 3.

> __Online resources__:
see Morgan Osborne paper.

**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
3^{2} − 2^{3}
= 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
*p*^{2}) and *q*^{p−1}
= 1 (mod *q*^{2}); 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 ×
10^{6}; 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 S^{n}
is S^{n} 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 NYT(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; 2018, Michael
Atiyah 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];
news sn(19)may [possible progress].

@ __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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