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

1. Kaplansky, Irving (2003), "The forms x + 32y² and x + 64y^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 x² + ny², 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.