On continued fraction partial quotients of square roots of primes

TL;DR

This paper investigates the patterns of partial quotients in the continued fraction expansions of square roots of primes, establishing finiteness results, divisibility conditions, and bounds related to their periods.

Contribution

It provides new finiteness results, divisibility criteria, and bounds for period lengths of continued fractions of square roots of primes and related integers.

Findings

Finitely many primes have a given partial quotient appearing an odd number of times in their continued fraction periods.

If the period length of sqrt(D) is divisible by 4, then 1 appears as a partial quotient.

An upper bound for the period length of continued fractions of sqrt(D) is established.

Abstract

We show that for each positive integer $a$ there exist only finitely many prime numbers $p$ such that $a$ appears an odd number of times in the period of continued fraction of $\sqrt{p}$ or $\sqrt{2p}$. We also prove that if $p$ is a prime number and $D=p$ or $2p$ is such that the length of the period of continued fraction expansion of $\sqrt{D}$ is divisible by $4$, then $1$ appears as a partial quotient in the continued fraction of $\sqrt{D}$. Furthermore, we give an upper bound for the period length of continued fraction expansion of $\sqrt{D}$, where $D$ is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas.