On the period of the continued fraction of $\sqrt{pq}$
Volume 196 / 2020
Acta Arithmetica 196 (2020), 291-302
MSC: Primary 11A55; Secondary 11R11, 11R27, 11R29.
DOI: 10.4064/aa190828-5-12
Published online: 6 August 2020
Abstract
We consider the period of the regular continued fraction of $\sqrt {pq}$ where $p \lt q$ are two primes congruent to $3$ modulo $4$. We show that the length of the period is divisible by $4$ when $q$ is a quadratic non-residue modulo $p$ and is of the form $4k+2$ when $q$ is a quadratic residue modulo $p$. We also examine the parity of the central term in the palindromic part of the period.
