On polynomial-factorial diophantine equations

Daniel Berend, Jørgen E. Harmse

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

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 xr(x + 1) = n! has only finitely many solutions for r ≥ 4, but not that this is the case for 1 ≤ r ≤ 3 (although it undoubtedly should be). We also study the equation P(x) = Hn, where (Hn) is one of several other "highly divisible" sequences, proving again that for various classes of polynomials these equations have only finitely many solutions.

Original languageEnglish
Pages (from-to)1741-1779
Number of pages39
JournalTransactions of the American Mathematical Society
Volume358
Issue number4
DOIs
StatePublished - 1 Apr 2006

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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