International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 33, Pages 1737-1746

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

Fadwa S. Abu Muriefah and Amal Al-Rashed

Mathematics Department, Girls College of Education, P.O. Box 60561, Riyadh 11555, Saudi Arabia

Received 15 May 2003

Copyright © 2004 Fadwa S. Abu Muriefah and Amal Al-Rashed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 z2ckx2=ck1, 5z2cky2=ck5 are given by (x,y,z)=(0,±2,±sk), is valid for 2k31, then it is valid for all positive integer k.