#1802 · Number Theory · Diophantine equations

Sum of Convergents for Square Root

If $\frac{p _{n}}{q _{n}}$ be the $n^{th}$ convergent to $a^{2} + 1$ , show that $\frac{p _{2}^{2} + p _{3}^{2} + \dots + p _{n + 1}^{2}}{q _{2}^{2} + q _{3}^{2} + \dots + q _{n + 1}^{2}} = \frac{p _{n + 1} p _{n + 2} - p _{1} p _{2}}{q _{n + 1} q _{n + 2} - q _{1} q _{2}}$ .

~20 minProof · checked by AIcontinued fractionsconvergents

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXVII. Recurring Continued Fractions, Examples XXVII.b, No. 14

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