# On consecutive 1's in continued fractions expansions of square roots of prime numbers

## TL;DR

This paper investigates the occurrence of consecutive 1's in the continued fraction expansions of square roots of primes, proving their frequent existence and providing computational and conjectural insights.

## Contribution

It establishes unconditionally that many primes have continued fractions with three initial consecutive 1's, and explores related computational and theoretical questions.

## Key findings

- At least N log^{-3/2} N primes ≤ N have √p with three initial consecutive 1's
- Computational results support the theoretical findings
- Formulation of open problems and conjectures under Hypothesis H

## 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 $N\log^{-3/2}N$ prime numbers $p\leq N$ 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.