On continued fraction partial quotients of square roots of primes

Abstract

We show that for each positive integer aa there exist only finitely many prime numbers pp such that aa appears an odd number of times in the period of continued fraction of p\sqrt{p} or 2p\sqrt{2p}. We also prove that if pp is a prime number and D=pD=p or 2p2p is such that the length of the period of continued fraction expansion of D\sqrt{D} is divisible by 44, then 11 appears as a partial quotient in the continued fraction of D\sqrt{D}. Furthermore, we give an upper bound for the period length of continued fraction expansion of D\sqrt{D}, where DD 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.Comment: 14 pages, to appear in JN

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