Continued fractions with non-integer numerators

John R Greene, Jesse Schmieg

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


Anselm and Weintraub investigated a generalization of classic continued fractions, where the “numerator” 1 is replaced by an arbitrary positive integer. Here, we gener- alize further to the case of an arbitrary real number z ≥ 1. We focus mostly on the case where z is rational but not an integer. Extensive attention is given to periodic expansions and expansions for √n, where we note similarities and differences between the case where z is an integer and when z is rational. When z is not an integer, it need no longer be the case that √n has a periodic expansion. We give several infinite families where periodic expansions of various types exist.

Original languageEnglish (US)
Article number17.1.2
JournalJournal of Integer Sequences
Issue number1
StatePublished - Jan 1 2017


  • Continued fraction
  • Linear diophantine equation
  • Pell’s equation


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