On polynomial-factorial diophantine equations

Berend, Daniel; Harmse, Jørgen

doi:10.1090/S0002-9947-05-03780-3

On polynomial-factorial diophantine equations
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by Daniel Berend and Jørgen E. Harmse PDF

Trans. Amer. Math. Soc. 358 (2006), 1741-1779 Request permission

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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