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All cases of the form (2, 3, n) or (2, n, 3) have the solution 2 3 + 1 n = 3 2 which is referred below as the Catalan solution. The case x = y = z ≥ 3 is Fermat's Last Theorem , proven to have no solutions by Andrew Wiles in 1994.
The Beal conjecture, also known as the Mauldin conjecture [162] and the Tijdeman-Zagier conjecture, [163] [164] [165] states that there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k being greater than 2.
⇐generalized Riemann hypothesis [2] ⇐Selberg conjecture B [3] Emil Artin: 325 Bateman–Horn conjecture: number theory: Paul T. Bateman and Roger Horn: 245 Baum–Connes conjecture: operator K-theory: ⇒Gromov-Lawson-Rosenberg conjecture [4] ⇒Kaplansky-Kadison conjecture [4] ⇒Novikov conjecture [4] Paul Baum and Alain Connes: 2670 Beal ...
"Any solutions to the Beal conjecture will necessarily involve three terms all of which are ...." It is unfortunate to say a "solution" is to "the Beal conjecture". Each "solution" referred to here is a point (A,B,C) of the locus {(A,B,C) ∈ ℕ 3 | A x + B y = C z}. It is entirely correct to say that (A,B,C) is a solution to the equation A x ...
Schanuel's conjecture; Schinzel's hypothesis H; Scholz conjecture; Second Hardy–Littlewood conjecture; Serre's conjecture II; Sexy prime; Sierpiński number; Singmaster's conjecture; Safe and Sophie Germain primes; Stark conjectures; Sums of three cubes; Superperfect number; Supersingular prime (algebraic number theory) Szpiro's conjecture
The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r , then the L -function L ( E , s ) associated with it vanishes to order r at s = 1 .