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On the period of the continued fraction of $\sqrt{pq}$

Volume 196 / 2020

Shamik Das, Debopam Chakraborty, Anupam Saikia 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.

Authors

  • Shamik DasDepartment of Mathematics
    Indian Institute of Technology Guwahati
    Guwahati 781039, Assam, India
    e-mail
  • Debopam ChakrabortyDepartment of Mathematics
    BITS Pilani
    Hyderabad 500078, India
    e-mail
  • Anupam SaikiaDepartment of Mathematics
    Indian Institute of Technology Guwahati
    Guwahati 781039, Assam, India
    e-mail

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