#1799 · Number Theory · Diophantine equations

Relation among periodic continued fractions

If $x = \frac{1}{a _{1} +} \frac{1}{a _{2} +} \frac{1}{a _{1} +} \frac{1}{a _{2} +} \dots$ , $y = \frac{1}{2 a _{1} +} \frac{1}{2 a _{2} +} \frac{1}{2 a _{1} +} \frac{1}{2 a _{2} +} \dots$ , and $z = \frac{1}{3 a _{1} +} \frac{1}{3 a _{2} +} \frac{1}{3 a _{1} +} \frac{1}{3 a _{2} +} \dots$ , shew that $x (y^{2} - z^{2}) + 2 y (z^{2} - x^{2}) + 3 z (x^{2} - y^{2}) = 0$ .

~25 minProof · checked by AIcontinued fractionsalgebraic identity

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXVII. Recurring Continued Fractions, Examples XXVII.b, No. 11

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