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]

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.^[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 + 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 is congruent to 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 (since 113 = 9² + 32×1² and 113 = 7² + 64×1²).
The prime p = 41 is congruent to 9 modulo 16 and is representable by x² + 32y² (since 41 = 3² + 32×1²), but not by x² + 64y².
The prime p = 73 is congruent to 9 modulo 16 and is representable by x² + 64y² (since 73 = 3² + 64×1²), but not by x² + 32y².

Similar results

Five results similar to Kaplansky's theorem are known:^[3]

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 or 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 or 25 modulo 39 is representable by exactly one of these quadratic forms.
A prime p congruent to 1, 16, 26, 31 or 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 or 49 modulo 55 is representable by exactly one of these quadratic forms.
A prime p congruent to 1, 65 or 81 modulo 112 is representable by both or none of x² + 14y² and x² + 448y², whereas a prime p congruent to 9, 25 or 57 modulo 112 is representable by exactly one of these quadratic forms.
A prime p congruent to 1 or 169 modulo 240 is representable by both or none of x² + 150y² and x² + 960y², whereas a prime p congruent to 49 or 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 $x + 32 y 2 and x + 64 y^2$ [sic]", Proceedings of the 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 $x 2 + ny 2$ , New York: John Wiley & Sons, ISBN 0-471-50654-0, MR 1028322.
^ Brink, David (2009), "Five peculiar theorems on simultaneous representation of primes by quadratic forms", Journal of Number Theory, 129 (2): 464–468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.

[1] Kaplansky, Irving (2003), "The forms $x + 32 y 2 and x + 64 y^2$ [sic]", Proceedings of the American Mathematical Society, 131 (7): 2299–2300 (electronic), doi:10.1090/S0002-9939-03-07022-9, MR 1963780.

[2] Cox, David A. (1989), Primes of the form $x 2 + ny 2$ , New York: John Wiley & Sons, ISBN 0-471-50654-0, MR 1028322.

[3] Brink, David (2009), "Five peculiar theorems on simultaneous representation of primes by quadratic forms", Journal of Number Theory, 129 (2): 464–468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.

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