On the extendibility of the Diophantine triple {1,5,c}

Abstract

We study the problem of extendibility of the triples of the form {1,5,c}. We prove that if ck=sk2+1, where (sk) is a binary recursive sequence, k is a positive integer, and the statement that all solutions of a system of simultaneous Pellian equations z2−ckx2=ck−1, 5z2−cky2=ck−5 are given by (x,y,z)=(0,±2,±sk), is valid for 2≤k≤31, then it is valid for all positive integer k.