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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Zhenfu Cao
Journal: Proc. Amer. Math. Soc. 128 (2000), 1927-1931
MSC (2000): Primary 11D61, 11D41
Published electronically: February 25, 2000
MathSciNet review: 1694856
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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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Additional Information

Zhenfu Cao
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
Email: zfcao@hope.hit.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05517-9
PII: S 0002-9939(00)05517-9
Keywords: Exponential Diophantine equation, higher degree Diophantine equation, Adachi's theorem, Pell's equation, Bernoulli number
Received by editor(s): September 8, 1998
Published electronically: February 25, 2000
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2000 American Mathematical Society