Kaplansky's theorem on quadratic forms, proved by Irving Kaplansky and published in 2003 in the Proceedings of the American Mathematical Society, concerns which primes are represented by the quadratic forms x squared plus 32 y squared and x squared plus 64 y squared. It states that a prime congruent to 1 modulo 16 is represented by both of these forms or by neither, while a prime congruent to 9 modulo 16 is represented by exactly one of them. No single congruence condition determines which primes each form represents on its own, yet the pairing of the two forms produces this precise and predictable pattern. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementA prime p congruent to 1 modulo 16 is representable by both or none of x^2 + 32y^2 and x^2 + 64y^2, whereas a prime p congruent to 9 modulo 16 is representable by exactly one of them. 1 Classification
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Source Kaplansky's theorem on quadratic forms (Wikipedia)
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1. Kaplansky's theorem on quadratic forms (Wikipedia)
Statement of the theorem
a prime p congruent to 1 modulo 16 is representable by both or none of x2 + 32y2 and x2 + 64y2, whereas a prime p congruent to 9 modulo 16 is representable by exactly one of these quadratic forms
Introduction, second sentence
It was proved in 2003 by Irving Kaplansky.
In Branch: Number Theory, Lead section
In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of
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