The diophantine equation 8x + py = z2

Abstract

Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ± 3 (mod 8), then the equation 8x + py = z2 has no positive integer solutions (x, y, z); (ii) if p ≡ 7 (mod 8), then the equation has only the solutions (p, x, y, z) = (2q - 1, (1 / 3) (q + 2), 2, 2q + 1), where q is an odd prime with q ≡ 1 (mod 3); (iii) if p ≡ 1 (mod 8) and p ≠ 17, then the equation has at most two positive integer solutions (x, y, z).