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 2⌊√*d*⌋. 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 2⌊√26⌋ = 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.