#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

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