ON THE EXPONENTIAL DIOPHANTINE EQUATION px − 2y = z2 WITH p = k2 + 2, A PRIME NUMBER

Abstract

We are interested in finding non-negative integer solutions for the Diophantine equation px − 2y = z2, where p = k2 + 2 is a prime number and k ≥ 0. We show that all the positive integer solutions of this equation are given by (1, 1, k) if p ≥ 11, (1, 1, 1), (3, 1, 5), (2, 3, 1) if p = 3. In the case p = 2 the equation has two infinite and disjoint families of solutions. The proofs are based on the use of the Catalan-Mihˇailescu Theorem (old Catalan conjecture) and properties of the modular arithmetic. In addition, we prove that equations of type px − 2y = w2u with u ≥ 2 do not have positive integer solutions if p ≥ 11 and k is not a perfect square. Moreover, we find exactly two positive integer solutions for px − 2y = w2u , with u ≥ 2, when p = 3.