Continued fractions with period 1

A while back I wrote about continued fractions of square roots. That post cited a theorem that if d is not a perfect square, then the continued fraction representation of d is periodic. The period consists of a palindrome followed by 2d. See that post for details and examples.

One thing the post did not address is the length of the period. The post gave the example that the continued fraction for 5 has period 1, i.e. the palindrome part is empty.

There's a theorem [1] that says this pattern happens if and only if d = n^2 + 1. That is, the continued fraction for d is periodic with period 1 if and only if d is one more than a square. So if we wanted to find the continued fraction expression for 26, we know it would have period 1. And because each period ends in 226 = 10, we know all the coefficients after the initial 5 are equal to 10.

[1] Samuel S. Wagstaff, Jr. The Joy of Factoring. Theorem 6.15.

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