## Abstract

Let K be a field of characteristic 0 and let (G_{n}(x)) _{n=0}^{∞} be a linear recurring sequence of degree d in K[x] defined by the initial terms G_{0}, ..., G_{d-1} ∈ K[x] and by the difference equation G_{n+d}(x) = A _{d-1}(x)G_{n+d-1}(x) + ... + A_{0}(x)G_{n}(x), for n ≥ 0, with A_{0}, ..., A_{d-1} ∈ K [x]. Finally, let P(x) be an element of K[x]. In this paper we are giving fairly general conditions depending only on G_{0},..., G_{d-1} on P, and on A_{0}, ..., A_{d-1} under which the Diophantine equation G _{n}(x) = G_{m}(P(x)) has only finitely many solutions (n, m) ∈ ℤ^{2}, n, m ≥ 0. Moreover, we are giving an upper bound for the number of solutions, which depends only on d. This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

Original language | English |
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Pages (from-to) | 4657-4681 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society |

Volume | 355 |

Issue number | 11 |

DOIs | |

Publication status | Published - 1 Nov 2003 |

## Keywords

- Diophantine equations
- Linear recurring sequences
- S-unit equations

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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