Journal of Applied Mathematics & Bioinformatics

A new proof of Eulerís theorem on Catalanís equation

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  •                                                Abstract

    This paper contains a new proof of Eulerís theorem, that the only non-trivial integral solution, (Š,‚), of Š2 = + 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 Eulerís isolated and other known proofs of this result, is that it charts a common path to a novel approach to the solution of Catalanís conjecture and indeed of any Diophantine equation.