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Genus Theorem


The Diophantine equation

 x^2+y^2=p

can be solved for p a prime iff p=1 (mod 4) or p=2. The representation is unique except for changes of sign or rearrangements of x and y. This theorem is intimately connected with the quadratic reciprocity theorem, and generalizes to the biquadratic reciprocity theorem.


See also

Composition Theorem, Diophantine Equation--4th Powers, Fermat's Theorem, Form Genus, Fundamental Theorem of Genera, Quadratic Reciprocity Theorem

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Cite this as:

Weisstein, Eric W. "Genus Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GenusTheorem.html

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