## A Simple Proof for Catalan's Conjecture

**Authors:** Julian TP Beauchamp

Catalan's Conjecture was first made by Belgian mathematician Eugène Charles Catalan in 1844, and states that 8 and 9 (2^3 and 3^2) are the only consecutive powers, excluding 0 and 1. That is to say, that the only solution in the natural numbers of a^x - b^y=1 for a,b,x,y > 1 is a=3, x=2, b=2, y=3. In other words, Catalan conjectured that 3^2-2^3=1 is the only nontrivial solution. It was finally proved in 2002 by number theorist Preda Mihailescu making extensive use of the theory of cyclotomic fields and Galois modules.

**Comments:** 6 Pages.

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### Submission history

[v1] 2019-06-22 06:25:55

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