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.[1] 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.[2] 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:[3] 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