## 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 ≥ 4, but not that this is the case for 1 ≤ r ≤ 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.

Original language | English |
---|---|

Pages (from-to) | 1741-1779 |

Number of pages | 39 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 4 |

DOIs | |

State | Published - 1 Apr 2006 |

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics