#1798 · Number Theory · Diophantine equations

Convergents of the square root of a sum

If $a^{2} + 1$ be expressed as a continued fraction, shew that $2 (a^{2} + 1) q_{n} = p_{n - 1} + p_{n + 1}$ and $2 p_{n} = q_{n - 1} + q_{n + 1}$ .

~25 minProof · checked by AIcontinued fractionsconvergents

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXVII. Recurring Continued Fractions, Examples XXVII.b, No. 10

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