Full powers in arithmetic progressions (Q2714314)

The authors investigate the arithmetic progressions \(a^2, y^n, x^2,\) where \(x, y\) are coprime positive integers, while \(a\) and \(n\) are given integers with \(a>0\) and \(n\geq 3.\) The theorems of the paper provide upper bounds for the solutions of the Diophantine equation \(x^2+a^2=2y^n,\) for \(n\) and for the number of the solutions. As a numerical result, all solutions of the equation \(x^2+a^2=2y^n\) are listed in the case when \(3\leq n\leq 80\) and \(1\leq a\leq 1000\).