© Hindawi Publishing Corp. ON THE EXTENDIBILITY OF THE DIOPHANTINE TRIPLE {1,5,c}

We study the problem of extendibility of the triples of the form {1,5,c}. We prove that if ck = s2k + 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. 2000 Mathematics Subject Classification: 11D09, 11D25