The diophantine equation x 2 + 2 a · 17 b = y n

Abstract

Let ℤ, ℕ be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, there have been many papers concerned with solutions (x, y, n, a, b) of the equation x 2 + 2a p b = y n, x, y, n ε ℕ, gcd(x, y) = 1, n ⩾ 3, a, b ε ℂ, a ⩾ 0, b ⩾ 0. And all solutions of it have been determined for the cases p = 3, p = 5, p = 11 and p = 13. In this paper, we mainly concentrate on the case p = 3, and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions (x, y, n, a, b) of the equation x 2+2a · 17b = y n, x, y, n ε ℙ, gcd(x, y) = 1, n ⩾ 3, a, b ∈ ℙ, a ⩾ 0, b ⩾ 0, are determined.