(Q17098379)

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Kaplansky's theorem on quadratic forms

theorem that a prime congruent to 1 modulo 16 is representable by either both or neither of the quadratic forms x²+32y² and x²+64y², while a prime congruent to 9 modulo 16 is representable by exactly one of the two

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defining formula

{\begin{aligned}p\equiv 1{\pmod {16}}&\implies \left(\left(\exists x,y\in \mathbb {Z} \colon p=x^{2}+32y^{2}\right)\iff \left(\exists x,y\in \mathbb {Z} \colon p=x^{2}+64y^{2}\right)\right)\\p\equiv 9{\pmod {16}}&\implies \left(\left(\exists x,y\in \mathbb {Z} \colon p=x^{2}+32y^{2}\right)\iff \left(\nexists x,y\in \mathbb {Z} \colon p=x^{2}+64y^{2}\right)\right)\end{aligned}}

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proved by

Irving Kaplansky

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maintained by WikiProject

WikiProject Mathematics

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/m/05b1dwb

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enwiki Kaplansky's theorem on quadratic forms

(Q17098379)

Kaplansky's theorem on quadratic forms

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