Even and odd periods in continued fractions of square roots

Rippon, Philip and Taylor, Harold (2004). Even and odd periods in continued fractions of square roots. Fibonacci Quarterly, 42(2) pp. 170–180.

URL: http://www.engineering.sdstate.edu/~fib/fibpreviou...

Abstract

The continued fraction for $\sqrt{N}$, where $N$ is a positive integer, has the periodic form
$\sqrt{N}=[a_0,\overline{a_1,a_2,\ldots, a_l}\,],$
where $a_1,a_2,\ldots,a_{l-1}$ is a palindrome and $a_l=2a_0$. The period $l=l(N)$ is assumed to be
of minimal length. We give several new results concerning the intriguing question: How can we distinguish between those integers $N$ for which $l(N)$ is even and
those for which $l(N)$ is odd?

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