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arXiv · 1406.7571

End-symmetric continued fractions and quadratic congruences

Barry R. Smith
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Abstract

We show that for a fixed integer $n \neq \pm2$, the congruence $x^2 + nx \pm 1 \equiv 0 \pmodα$ has the solution $β$ with $0 < β< α$ if and only if $α/β$ has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number $α/β> 1$ in lowest terms has a symmetric continued fraction precisely when $β^2 \equiv \pm 1 \pmodα$.

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Barry R. Smith. 2014-12-08. End-symmetric continued fractions and quadratic congruences. https://arxiv.org/abs/1406.7571

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