Kaplansky's theorem on quadratic forms

• 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.

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.