On the power values of the sum of three squares in arithmetic progression

Abstract

In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation (x-d)(2) +x(2)+ (x+d)(2) = y(n) (*), when n is an odd prime and d = p(r), p > 3, a prime. So this improves the results of the papers of A. Koutsianas and V. Patel [19] and A. Koutsianas [18]. Secondly, under the assumption of our first result, we prove that (*) has at most one solution (x, y). Next, for a general d, we prove the following two results: (i) if every odd prime divisor q of d satisfies q +/- 1 (mod 2n), then (*) has only the solution (x, y, d, n) = (21, 11, 2, 3), and (ii) if n > 228000 and d > 8 root 2, then all solutions (x,y) of (*) satisfy y(n) < 2(3/2)d(3).