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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On polynomial-factorial diophantine equations


Authors: Daniel Berend and Jørgen E. Harmse
Journal: Trans. Amer. Math. Soc. 358 (2006), 1741-1779
MSC (2000): Primary 11D99; Secondary 11B65
Published electronically: October 21, 2005
MathSciNet review: 2186995
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Abstract: We study equations of the form $P(x)=n!$ and show that for some classes of polynomials $P$ the equation has only finitely many solutions. This is the case, say, if $P$ is irreducible (of degree greater than 1) or has an irreducible factor of ``relatively large" degree. This is also the case if the factorization of $P$ contains some ``large" power(s) of irreducible(s). For example, we can show that the equation $x^{r}(x+1)=n!$ has only finitely many solutions for $r\ge 4$, but not that this is the case for $1\le r\le 3$ (although it undoubtedly should be). We also study the equation $P(x)=H_{n}$, where $(H_{n})$ is one of several other ``highly divisible" sequences, proving again that for various classes of polynomials these equations have only finitely many solutions.


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Additional Information

Daniel Berend
Affiliation: Departments of Mathematics and of Computer Science, Ben-Gurion University, Beer-Sheva 84105, Israel

Jørgen E. Harmse
Affiliation: Analysis and Applied Research Division, BAE Systems, Building 27-16, 6500 Tracor Lane, Austin, Texas 78725

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03780-3
PII: S 0002-9947(05)03780-3
Received by editor(s): July 10, 2002
Received by editor(s) in revised form: July 9, 2004
Published electronically: October 21, 2005
Additional Notes: The first author’s research was supported in part by the Israel Science Foundation (Grant #186/01)
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.