Genus Theorem

The Diophantine equation


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.

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