Is there any known complete parametrization of the Diophantine equation $$ A^{2} + B^{2} = C^{2} + D^{2} $$ where $A, B, C, D$ are (positive) rational numbers, or equivalently, integers?
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1https://math.stackexchange.com/questions/153603/diophantine-equation-a2b2-c2d2/736164#736164 – individ Mar 08 '19 at 17:53
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Yes, there are. You can put $$A=ms+nt\\B=nt-ms\\C=ms-nt\\D=mt+ns$$ where $m,n,s,t$ are arbitrary.
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In your "soloution" it seems impossible for both B and C to be positive. – Peter Green Mar 08 '19 at 18:41
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It should be $(A,B,C,D) = (p r + q s , q r - p s , p r - q s , p s + q r),$ – Dietrich Burde Mar 08 '19 at 19:40
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@Peter Green: It is not "my solution" but a quite known parameterization in which I have had an obvious lapse. – Piquito Mar 09 '19 at 19:39
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