A NOTE ON THE DIOPHANTINE EQUATION

Abstract

AbstractLet $q$ be an odd prime such that ${q}^{t} + 1= 2{c}^{s} $, where $c, t$ are positive integers and $s= 1, 2$. We show that the Diophantine equation ${x}^{2} + {q}^{m} = {c}^{n} $ has only the positive integer solution $(x, m, n)= ({c}^{s} - 1, t, 2s)$ under some conditions. The proof is based on elementary methods and a result concerning the Diophantine equation $({x}^{n} - 1)/ (x- 1)= {y}^{2} $ due to Ljunggren. We also verify that when $2\leq c\leq 30$ with $c\not = 12, 24$, the Diophantine equation ${x}^{2} + \mathop{(2c- 1)}\nolimits ^{m} = {c}^{n} $ has only the positive integer solution $(x, m, n)= (c- 1, 1, 2). $