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On polynomial-factorial diophantine equations

Authors:	Daniel Berend Jø rgen E Harmse

Institution:	Departments of Mathematics and of Computer Science, Ben-Gurion University, Beer-Sheva 84105, Israel Jørgen E. Harmse ; Analysis and Applied Research Division, BAE Systems, Building 27-16, 6500 Tracor Lane, Austin, Texas 78725

Abstract:	We study equations of the form and show that for some classes of polynomials the equation has only finitely many solutions. This is the case, say, if 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 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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