On Consecutive 1’s in Continued Fractions Expansions of Square Roots of Prime Numbers

Abstract

In this note, we study the problem of existence of sequences of consecutive 1’s in the periodic part of the continued fractions expansions of square roots of primes. We prove unconditionally that, for a given $N\gg 1,$ there are at least $N{\hspace{0.17em}\text{log}\hspace{0.17em}}^{-3/2}N$ prime numbers $p\le N$ such that the continued fraction expansion of $\sqrt{p}$ contains three consecutive 1’s on the beginning of the periodic part. We also present results of our computations related to the considered problem and some related problems, formulate several open questions and conjectures and get some results under the assumption of Hypothesis H of Schinzel.

Keywords:

• prime numbers
• continued fractions
• period
• experimental mathematics
• numerical computation

Acknowledgments

We are grateful for remarks of prof. Schinzel concerning the initial form of the manuscript. We thank also the anonymous referee for careful reading of the manuscript and useful remarks improving the presentation.

Additional information

Funding

The research of the authors was supported by the Polish National Science Centre grants: UMO-2018/29/N/ST1/00470 (PM) and UMO-2012/07/E/ST1/00185 (MU)