English

Abstract

Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 ∤ x, p = ±3 (mod 8), the Diophantine equation (2x − 1)(py − 1) = 2z2 has no positive integer solution except when p = 3 or p is of the form $$p = 2a_0^2 + 1$$ , where a0 > 1 is an odd positive integer. (2) if 2 ∤ x, 2 ∣; y, y ≠ 2, 4, then the Diophantine equation (2x − 1)(py − 1) = 2z2 has no positive integer solution.