On generalized continued fractions converging to an irrational number

Hayato Yoshida
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 622–628
DOI: 10.7546/nntdm.2026.32.3.622-628
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Authors and affiliations

Hayato Yoshida
Mathematics, Integrated Science and Engineering Major, Graduate School of Science and Engineering, Kansai University
3-3-35, Yamate-cho, Osaka, Suita, 564-8680, Japan

Abstract

In [5], we give a sufficient condition for generalized continued fractions to converge to an irrational number. Specifically, we determine how many non-integers must exist in c_{1},c_{2},\ldots, c_{i} in order for the continued fraction [0; c_{1},c_{2},\ldots] to be irrational, where c_{i} are rational numbers greater than M>1 (denoted by c_{i}\in \mathbb{Q}_{\ge M}) and the denominator of c_{i} less than or equal to positive integer N. In [6], we extended this result from the case M>1 to M>0. In this paper, we give a further extension by removing the restrictions on the denominators of the partial quotients. Let M be a real number greater than 0. We consider continued fractions [0;c_{1},c_{2},\ldots], where c_{i} are rational numbers greater than or equal to M for any positive integer i. We give a sufficient condition for it to converge to an irrational number and some interesting examples.

Keywords

  • Continued fractions
  • Irrational numbers
  • Diophantine approximations

2020 Mathematics Subject Classification

  • 11A55

References

  1. Khinchin, A. Y. (1949). Continued Fractions (2nd ed.). Gosudarstvennoe Izdatel’stvo Tekhniko-Teoreticheskoi Literatury. Moscow–Leningrad (in Russian); English translation, University of Chicago Press, Chicago-London, 1964.
  2. Legendre, A. M. (1794). Éléments de Géométrie. Firmin Didot, Paris.
  3. Lorentzen, L., & Waadeland, H. (1992). Continued Fractions with Applications. North-Holland.
  4. Perron, O. (1913). Die Lehre von den Kettenbrüchen. B.G. Teubner, Leipzig.
  5. Yoshida, H. (2025). A simple condition for generalized continued fractions to converge to an irrational number. Research in Number Theory, 11, Article ID 36.
  6. Yoshida, H. (2025). An extension of conditions for generalized continued fractions to converge to an irrational number. The Ramanujan Journal, 68, Article ID 73.

Manuscript history

  • Received: 31 January 2026
  • Revised: 25 May 2026
  • Accepted: 29 October 2026
  • Online First: 31 October 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Yoshida, H. (2026). On generalized continued fractions converging to an irrational number. Notes on Number Theory and Discrete Mathematics, 32(3), 622-628, DOI: 10.7546/nntdm.2026.32.3.622-628.

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