#1801 · Number Theory · Diophantine equations

Periodic Continued Fractions Identity

If $x = a + \frac{1}{b + \frac{1}{a + \frac{1}{b + \frac{1}{a + \dots}}}}$ and $y = b + \frac{1}{a + \frac{1}{a + \frac{1}{b + \frac{1}{b + \dots}}}}$ , show that $(a b^{2} + a + b) x - (a^{2} b + a + b) y = a^{2} - b^{2}$ .

~15 minProof · checked by AIcontinued fractionsinfinite fractions

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXVII. Recurring Continued Fractions, Examples XXVII.b, No. 13

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