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On the Diophantine equation
Authors:Zhenfu Cao
Institution:Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People's Republic of China
Abstract:Let $p$ be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if $p \equiv 1(\text{mod }4)$ and $p\nmid B_{(p-1)/2}$, then the equation $ x^{p}+1=py^{2},\,\,y\ne 0$, and the equation $x^{p}+2^{2m}=py^{2},\,\,m\in \mathbb{N} , \text{ gcd}(x, y )=1,\,\, p\mid y$, have no integral solutions respectively. Here $B_{(p-1)/2} $ is $(p-1)/2$th Bernoulli number.

Keywords:Exponential Diophantine equation  higher degree Diophantine equation  Adachi's theorem  Pell's equation  Bernoulli number
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