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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.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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 ...
In 1993, he publicly stated a new conjecture, known as the Beal Conjecture, that implies Fermat's Last Theorem as a corollary. No counterexample has been found to the conjecture. To encourage research on the conjecture, Beal has personally funded a standing prize of $1 million for its proof or disproof.
"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 ...