#1806 · Number Theory · Diophantine equations

Convergents at Period Lengths

If $N$ is converted into a continued fraction, and if $n$ is the number of quotients in the period, show that $q_{2 n} = 2 p_{n} q_{n}$ and $p_{2 n} = 2 p_{n}^{2} + (- 1)^{n + 1}$ .

~25 minProof · checked by AIcontinued fractionsquadratic surdspell's equation

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXVII. Recurring Continued Fractions, Examples XXVII.b, No. 18

Stuck? A gentle nudge

Something off?

Workspace

Your working

Check your full solution

Attach photos of your pages (up to 3) or one PDF — or check what you typed above. Lemma reads the work and verifies the reasoning, not just the answer.

Your first check today doesn't need an account.

Your photos and PDFs are read once and not stored — only the transcription is kept.