Complete solution of a problem of Diophantus and Euler
by
Clemens Fuchs
Graz University of Technology
Coauthors: Andrej Dujella

Diophantus and Euler studied the problem of finding sets of numbers with the property that the product of any two of its distinct elements plus their sum is a perfect square. The conjecture is that there does not exist a set of four positive integers with this property. In this talk I will first discuss how this problem is related to so-called Diophantine D(n)-m-tuples (that are sets with the property that the product of any two distinct elements plus n is a perfect square) and I will present a proof of the above conjecture. Among others the proof uses a theorem on simultaneous Diophantine approximation of square roots which are close to 1 due to Bennett.

Date received: April 26, 2004