On continued fraction partial quotients of square roots of primes

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 p or 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 D is divisible by 4, then 1 appears as a partial quotient in the continued fraction of D. Furthermore, we give an upper bound for the period length of continued fraction expansion of 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 [MU].