On consecutive perfect powers with elementary methods

2013/05/04 by Paolo Leonetti, Leonetti, Paolo
Mathematics · #08-02 #FOS: Mathematics #Number Theory (math.NT) #Primary 11D72 #Secondary 11A99 #math.NT #msc:08-02 #msc:11A99 #msc:11D72

paper · pdf · doi:10.48550/arxiv.1305.0892

Improved exposition and result in Section 9

arxiv created 2017/02/11 · arxiv updated 2017/02/14

Abstract

Catalan's conjecture claims that the Diophantine equation x^p-y^q=1 admits the unique solution 3²-2³=1 in integers x,y,p,q ≥ 2. The conjecture has been finally proved by P. Mihăilescu (2002) using the theory of cyclotomic fields and Galois modules. Here, relying only on elementary techniques, we prove several instances of this classical result. In particular, we prove the conjecture in the following cases: p even (due to V.A. Lebesgue), q is even (due to L. Euler and Chao Ko), x divides q, y divides x-1, y is a power of a prime, and y≤ p^p/2.

Related

• Catalan's Conjecture over Number Fields
• Catalan's Equation For Even Primes over Quadratic Fields
• Catalan's conjecture is Mihăilescu's theorem
• Differences between perfect powers : the Lebesgue-Nagell Equation
• Perfect powers in alternating sum of consecutive cubes