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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The Diophantine equation $ f(x)=g(y)$


Author: Todd Cochrane
Journal: Proc. Amer. Math. Soc. 109 (1990), 573-577
MSC: Primary 11D41
DOI: https://doi.org/10.1090/S0002-9939-1990-1019271-X
MathSciNet review: 1019271
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Abstract: Let $ f(x),g(y)$ be polynomials over $ \mathbb{Z}$ of degrees $ n$ and $ m$ respectively and with leading coefficients $ {a_n},{b_m}$. Suppose that $ m\vert n$ and that $ {a_n}/{b_m}$ is the $ m$th power of a rational number. We give two elementary proofs that the equation $ f(x) = g(y)$ has at most finitely many integral solutions unless $ f(x) = g(h(x))$ for some polynomial $ h(x)$ with rational coefficients taking integral values at infinitely many integers.


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DOI: https://doi.org/10.1090/S0002-9939-1990-1019271-X
Article copyright: © Copyright 1990 American Mathematical Society