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.

**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 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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