# Kaplansky's theorem on quadratic forms Kaplansky's theorem on quadratic forms In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Irving Kaplansky. Conteúdo 1 Declaração do teorema 2 Prova 3 Exemplos 4 Similar results 5 Notes Statement of the theorem Kaplansky's theorem states that a prime p congruent to 1 módulo 16 is representable by both or none of x2 + 32y2 and x2 + 64ano 2, whereas a prime p congruent to 9 módulo 16 is representable by exactly one of these quadratic forms.

This is remarkable since the primes represented by each of these forms individually are not describable by congruence conditions. Proof Kaplansky's proof uses the facts that 2 is a 4th power modulo p if and only if p is representable by x2 + 64ano 2, and that −4 is an 8th power modulo p if and only if p is representable by x2 + 32ano 2.

Examples The prime p = 17 é congruente com 1 módulo 16 and is representable by neither x2 + 32y2 nor x2 + 64ano 2. The prime p=113 is congruent to 1 módulo 16 and is representable by both x2 + 32y2 and x2+64y2 (desde 113 = 92 + 32×12 and 113 = 72 + 64×12). The prime p = 41 é congruente com 9 módulo 16 and is representable by x2 + 32ano 2 (desde 41 = 32 + 32×12), but not by x2 + 64ano 2. The prime p = 73 é congruente com 9 módulo 16 and is representable by x2 + 64ano 2 (desde 73 = 32 + 64×12), but not by x2 + 32ano 2. Similar results Five results similar to Kaplansky's theorem are known: A prime p congruent to 1 módulo 20 is representable by both or none of x2 + 20y2 and x2 + 100ano 2, whereas a prime p congruent to 9 módulo 20 is representable by exactly one of these quadratic forms. A prime p congruent to 1, 16 ou 22 módulo 39 is representable by both or none of x2 + xy + 10y2 and x2 + xy + 127ano 2, whereas a prime p congruent to 4, 10 ou 25 módulo 39 is representable by exactly one of these quadratic forms. A prime p congruent to 1, 16, 26, 31 ou 36 módulo 55 is representable by both or none of x2 + xy + 14y2 and x2 + xy + 69ano 2, whereas a prime p congruent to 4, 9, 14, 34 ou 49 módulo 55 is representable by exactly one of these quadratic forms. A prime p congruent to 1, 65 ou 81 módulo 112 is representable by both or none of x2 + 14y2 and x2 + 448ano 2, whereas a prime p congruent to 9, 25 ou 57 módulo 112 is representable by exactly one of these quadratic forms. A prime p congruent to 1 ou 169 módulo 240 is representable by both or none of x2 + 150y2 and x2 + 960ano 2, whereas a prime p congruent to 49 ou 121 módulo 240 is representable by exactly one of these quadratic forms.

It is conjectured that there are no other similar results involving definite forms.

Notes ^ Kaplansky, Irving (2003), "The forms x + 32y2 and x + 64y^2 [sic]", Anais da American Mathematical Society, 131 (7): 2299–2300 (electronic), doi:10.1090/S0002-9939-03-07022-9, MR 1963780. ^ Cox, David A. (1989), Primes of the form x2 + ny2, Nova york: John Wiley & Sons, ISBN 0-471-50654-0, MR 1028322. ^ Brink, Davi (2009), "Five peculiar theorems on simultaneous representation of primes by quadratic forms", Jornal da Teoria dos Números, 129 (2): 464-468, doi:10.1016/j.jnt.2008.04.007, MR 2473893. Categorias: Theorems in number theoryQuadratic forms

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