If-a-and-b-are-positive-integers-such-that-1-ab-divides-a-2-b-2-show-that-the-integer-a-2-b-2-1-ab-must-be-a-perfect-square-

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Question Number 56912 by Kunal12588 last updated on 26/Mar/19

If a and b are positive integers such that (1+ab) divides (a^2 +b^2 ) show that the integer ((a^2 +b^2 )/(1+ab)) must be a perfect square.

$$\mathrm{If}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\left(\mathrm{1}+{ab}\right)\:\mathrm{divides}\:\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{integer} \\ $$$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{\mathrm{1}+{ab}}\:\mathrm{must}\:\mathrm{be}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}. \\ $$

Commented by kaivan.ahmadi last updated on 26/Mar/19