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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 equations $ d\sb 1x\sp 2+2\sp {2m}d\sb 2=y\sp n$ and $ d\sb 1x\sp 2+d\sb 2=4y\sp n$


Author: Le Maohua
Journal: Proc. Amer. Math. Soc. 118 (1993), 67-70
MSC: Primary 11D61
MathSciNet review: 1152282
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Abstract: Let $ {d_1},\;{d_2}$ be coprime positive integers, which are squarefree, and let $ h$ denote the class number of the imaginary quadratic field $ \mathbb{Q}(\sqrt { - {d_1}{d_2}} )$. Let $ m,\;n$ be integers such that $ m \geqslant 0,\;n > 1$, and $ \gcd (n,2h) = 1$. In this paper we prove that if $ n \geqslant 8.5 \cdot {10^6}$, then the equations $ {d_1}{x^2} + {2^{2m}}{d_2} = {y^n}(2\nmid y)$ and $ {d_1}{x^2} + {d_2} = 4{y^n}$ have no positive integer solutions $ (x,y)$ with $ \gcd (x,y) = 1$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1152282-5
PII: S 0002-9939(1993)1152282-5
Article copyright: © Copyright 1993 American Mathematical Society