# 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. Contenuti 1 Enunciato del teorema 2 Prova 3 Esempi 4 Similar results 5 Notes Statement of the theorem Kaplansky's theorem states that 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.

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 + 64y2, and that −4 is an 8th power modulo p if and only if p is representable by x2 + 32y2.

Examples The prime p = 17 è congruente a 1 modulo 16 and is representable by neither x2 + 32y2 nor x2 + 64y2. The prime p=113 is congruent to 1 modulo 16 and is representable by both x2 + 32y2 and x2+64y2 (da 113 = 92 + 32×12 and 113 = 72 + 64×12). The prime p = 41 è congruente a 9 modulo 16 and is representable by x2 + 32y2 (da 41 = 32 + 32×12), but not by x2 + 64y2. The prime p = 73 è congruente a 9 modulo 16 and is representable by x2 + 64y2 (da 73 = 32 + 64×12), but not by x2 + 32y2. Similar results Five results similar to Kaplansky's theorem are known: A prime p congruent to 1 modulo 20 is representable by both or none of x2 + 20y2 and x2 + 100y2, whereas a prime p congruent to 9 modulo 20 is representable by exactly one of these quadratic forms. A prime p congruent to 1, 16 o 22 modulo 39 is representable by both or none of x2 + xy + 10y2 and x2 + xy + 127y2, whereas a prime p congruent to 4, 10 o 25 modulo 39 is representable by exactly one of these quadratic forms. A prime p congruent to 1, 16, 26, 31 o 36 modulo 55 is representable by both or none of x2 + xy + 14y2 and x2 + xy + 69y2, whereas a prime p congruent to 4, 9, 14, 34 o 49 modulo 55 is representable by exactly one of these quadratic forms. A prime p congruent to 1, 65 o 81 modulo 112 is representable by both or none of x2 + 14y2 and x2 + 448y2, whereas a prime p congruent to 9, 25 o 57 modulo 112 is representable by exactly one of these quadratic forms. A prime p congruent to 1 o 169 modulo 240 is representable by both or none of x2 + 150y2 and x2 + 960y2, whereas a prime p congruent to 49 o 121 modulo 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]", Atti dell'American Mathematical Society, 131 (7): 2299–2300 (electronic), doi:10.1090/S0002-9939-03-07022-9, SIG 1963780. ^ Cox, David A. (1989), Primes of the form x2 + ny2, New York: John Wiley & Sons, ISBN 0-471-50654-0, SIG 1028322. ^ Brink, Davide (2009), "Five peculiar theorems on simultaneous representation of primes by quadratic forms", Rivista di teoria dei numeri, 129 (2): 464–468, doi:10.1016/j.jnt.2008.04.007, SIG 2473893. Categorie: Theorems in number theoryQuadratic forms

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