How do you proof that the simply periodic continuous fraction is palindromic for the square root of positive primes?

Question

I have formulated this question based on the initial curiosity and further investigation of the topic posted here: Identity and possible generalization of the reflective periodic continued fractions

How do you proof that for the square root of any positive prime, the simply periodic continuous fraction is palindromic?

It should be true both for even:

$$\sqrt{Z^+_{prime}} = [a_0; a_1, a_2, ..., a_2, a_1, 2a_0]$$

and for odd:

$$\sqrt{Z^+_{prime}} = [a_0; a_1, ..., a_n, ..., a_1, 2a_0]$$

sequences.

See also: https://en.wikipedia.org/wiki/Square_root#As_periodic_continued_fractions

Answer 1

Here are five textbooks that deal with the palindromic feature of the continued fraction expansion of $\sqrt n$. Some give detailed proofs, some give it as an exercise with strong hints. All require reading some of the material leading up to the problem. There's just no really easy way – you have to roll your sleeves up and get to work!

1. Rosen, Elementary Number Theory, 4th edition, Section 12.4.

2. Roberts, Elementary Number Theory, Chapter XIII, problem 17, part vi (book includes complete solutions to all problems).

3. Shanks, Solved and Unsolved Problems in Number Theory, Exercise 138, page 186.

4. Steuding, Diophantine Analysis, Section 5.4.

5. Stark, An Introduction to Number Theory, Chapter 7, Miscellaneous Exercise 18.