On the Diophantine equation Gn(x) = Gm(P(x)): Higher-order recurrences

Clemens Fuchs*, Attila Petho, Robert F. Tichy

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let K be a field of characteristic 0 and let (Gn(x)) n=0 be a linear recurring sequence of degree d in K[x] defined by the initial terms G0, ..., Gd-1 ∈ K[x] and by the difference equation Gn+d(x) = A d-1(x)Gn+d-1(x) + ... + A0(x)Gn(x), for n ≥ 0, with A0, ..., Ad-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 G0,..., Gd-1 on P, and on A0, ..., Ad-1 under which the Diophantine equation G n(x) = Gm(P(x)) has only finitely many solutions (n, m) ∈ ℤ2, 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 languageEnglish
Pages (from-to)4657-4681
Number of pages25
JournalTransactions of the American Mathematical Society
Issue number11
Publication statusPublished - 1 Nov 2003


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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


Dive into the research topics of 'On the Diophantine equation Gn(x) = Gm(P(x)): Higher-order recurrences'. Together they form a unique fingerprint.

Cite this