The Diophantine equation 8(x) + p(y) = z(2).

Lan Qi,Xiaoxue Li

doi:10.1155/2015/306590

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 + p y = z 2 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).

Highlights

Let Z, N be the sets of all integers and positive integers, respectively
Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ±3(mod 8), the equation 8x + py = z2 has no positive integer solutions (x, y, z); (ii) if p ≡ 7(mod 8), 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, the equation has at most two positive integer solutions (x, y, z)
If p is an odd prime with p ≡ 1(mod 4), the equation u2 − pV2 = −1, u, V ∈ N

Summary

Introduction

The solutions (x, y, z) of the equation. In this paper, using certain results of exponential Diophantine equations, we prove a general result as follows. Where q is an odd prime with q ≡ 1(mod 3). If p ≡ 1(mod 8) and p ≠ 17, (1) has at most two solutions (x, y, z). The above theorem contains the results of [1, 2]. If p ≠ 17, (1) has at most one solution (x, y, z)

Preliminaries

Proof of Theorem