Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] H. Davenport, D. J. Lewis, and A. Schinzel, Equations of the form $ f(x) = g(y)$, Quart. J. Math. Oxford Ser. 2, 12 (1961), 304-312, MR 0137703 (25:1152)
  • [2] W. J. LeVeque, Studies in number theory, Math. Assoc. of Amer. 1969, pp. 4-24. MR 0250970 (40:4201)
  • [3] B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259. MR 828821 (88e:11050)
  • [4] G. Pólya and G. Szegö, Problems and theorems in analysis II, Springer-Verlag, New York, 1976.
  • [5] C. L. Siegel, Uber einige anwendungen diophantischer approximationen, Abh. Preuss. Akad. Wiss. Phys. - Mat. Kl., no. 1 (1929).
  • [6] T. A. Skolem, Videnskapsselskapets Skrifter, no. 17 (1921), Theorem 8.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11D41

Retrieve articles in all journals with MSC: 11D41


Additional Information

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

American Mathematical Society