Continued Fraction Expansions Towards Zaremba’s Conjecture

Abstract

In this article, we show that Zaremba’s conjecture holds for positive integers that appear as values of polynomials resulting from a recurrence formula and their powers of two. For example, Zaremba’s conjecture holds for all positive integers less than or equal to 100 and their powers of two. Also, as a special case, Zaremba’s conjecture holds for positive integers that appear as values of Fibonacci polynomials and powers of two.

KEYWORDS:

• continued fractions
• Zaremba’s conjecture
• folding Lemma
• Fibonacci polynomials

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 11A55
• Secondary 11J70
• 11B39

Acknowledgments

The authors thank the referee for carefully reading of the manuscript and for giving constructive comments.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Notes

1 In 1963 [3], another Fibonacci polynomial that becomes a Fibonacci number when X = 1/2 is introduced.