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.
.