Journal of Applied Mathematics & Bioinformatics

A new proof of Euler’s theorem on Catalan’s equation

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•                                                Abstract
  
  

  This paper contains a new proof of Euler’s theorem, that the only non-trivial integral solution, (α,β), of α² = β³ + 1 is  (+3,2). This proof employs only the properties of the ring, Z; of integers without recourse to elliptic curves and is independent of the methods of algebraic number fields. The advantage of our proof, over Euler’s isolated and other known proofs of this result, is that it charts a common path to a novel approach to the solution of Catalan’s conjecture and indeed of any Diophantine equation.