On the Diophantine equation 𝑥^{𝑝}+2^{2𝑚}=𝑝𝑦²

Cao, Zhenfu

doi:10.1090/S0002-9939-00-05517-9

On the Diophantine equation $x^{p}+2^{2m}=py^{2}$
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Proc. Amer. Math. Soc. 128 (2000), 1927-1931 Request permission

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.

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