#1807 · Number Theory · Diophantine equations

Periodicity of Surd Convergents

If $N$ be converted into a continued fraction, and if the penultimate convergents in the first, second, ... $k^{th}$ recurring periods be denoted by $n_{1}, n_{2}, \dots n_{k}$ respectively, show that $\frac{n_{k}+\sqrt{N}}{n_{k}-\sqrt{N}}=\left(\frac{n_{1}+\sqrt{N}}{n_{1}-\sqrt{N}} ight)^{k}$ .

~25 minProof · checked by AIcontinued fractionsrecurring continued fractionsconvergents

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXVII. Recurring Continued Fractions, Examples XXVII.b, No. 19

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