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 by Canadian mathematician Irving Kaplansky (1917-2006).[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.
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
- ^ See: I. Kaplansky, The forms x+32y2 and x+64y2 [sic], Procedings of the American Mathematical Society 131 (2003), no. 7, 2299--2300.
- ^ See: D. A. Cox, Primes of the Form x2+ny2, Wiley, New York, 1989.
- ^ See: D. Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, Journal of Number Theory 129 (2009), 464-468.