Fermat–Catalan conjecture

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In number theory, the Fermat–Catalan conjecture combines ideas of Fermat's last theorem and the Catalan conjecture, hence the name. The conjecture states that the equation

 

 

 

 

(1)

has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck); here a, b, c are positive coprime integers and m, n, k are positive integers satisfying

 

 

 

 

(2)

This inequality restriction on the exponents has the effect of precluding consideration of the known infinitude of solutions of (1) in which two of the exponents are 2 (such as Pythagorean triples).

As of 2015, the following ten solutions to (1) are known:[1]

The first of these (1m+23=32) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of (1) (since we can pick any m for m>6), these solutions only give a single triplet of values (am, bn, ck).

It is known by the Darmon–Granville theorem, which uses Faltings' theorem, that for any fixed choice of positive integers m, n and k satisfying (2), only finitely many coprime triples (abc) solving (1) exist;[2][3]:p. 64 but the full Fermat–Catalan conjecture is a much stronger statement since it allows for an infinitude of sets of exponents m, n and k.

The abc conjecture implies the Fermat–Catalan conjecture.[1]

Beal's conjecture is true if and only if all Fermat-Catalan solutions use 2 as an exponent for some variable.

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References[edit]

  1. ^ a b Pomerance, Carl (2008), "Computational Number Theory", in Gowers, Timothy; Barrow-Green, June; Leader, Imre, The Princeton Companion to Mathematics, Princeton University Press, pp. 361–362, ISBN 978-0-691-11880-2 .
  2. ^ Darmon, H.; Granville, A. (1995). "On the equations zm = F(x, y) and Axp + Byq = Czr". Bulletin of the London Mathematical Society. 27: 513–43. 
  3. ^ Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review. 1 (1). 

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