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 + 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.
- ↑ Cox, David A. (1989), Primes of the form x2 + ny2, 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.