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

摘要

This paper contains a new proof of Euler’s theorem, that the onlynon-trivial integral solution, (α,β), of α2= β3+1 is(+3,2). This proof employs only the properties of the ring, Z; of integers withoutrecourse to elliptic curves and is independent of the methods of algebraicnumber fields. The advantage of our proof, over Euler’s isolated and otherknown proofs of this result, is that it charts a common path to a novelapproach to the solution of Catalan’s conjecture and indeed of anyDiophantine equation.