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 = n² + 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.

.