On continued fraction partial quotients of square roots of primes

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 length of continued fraction expansion of '/ D. Furthermore, we give an upper bound for the period 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].& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).