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).

Statement of the theorem

Kaplansky's theorem states that a prime $p\equiv 1\mathrm {\ (mod\ } {16})$ is representable by both or none of $x^{2}+32y^{2}$ and $x^{2}+64y^{2}$, whereas a prime $p\equiv 9\mathrm {\ (mod\ } {16})$ 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.

Proof

Kaplansky's proof uses the facts that 2 is a 4th power modulo p if and only if p is representable by $x^{2}+64y^{2}$, and that -4 is an 8th power modulo p if and only if p is representable by $x^{2}+32y^{2}$.

Similar results

Five results similar to Kaplansky's theorem are known:

• A prime $p\equiv 1\mathrm {\ (mod\ } {20})$ is representable by both or none of $x^{2}+20y^{2}$ and $x^{2}+100y^{2}$, whereas a prime $p\equiv 9\mathrm {\ (mod\ } {20})$ is representable by exactly one of these forms.
• A prime $p\equiv 1,16,22\mathrm {\ (mod\ } {39})$ is representable by both or none of $x^{2}+xy+10y^{2}$ and $x^{2}+xy+127y^{2}$, whereas a prime $p\equiv 4,10,25\mathrm {\ (mod\ } {39})$ is representable by exactly one of these forms.
• A prime $p\equiv 1,16,26,31,36\mathrm {\ (mod\ } {55})$ is representable by both or none of $x^{2}+xy+14y^{2}$ and $x^{2}+xy+69y^{2}$, whereas a prime $p\equiv 4,9,14,34,49\mathrm {\ (mod\ } {55})$ is representable by exactly one of these forms.
• A prime $p\equiv 1,65,81\mathrm {\ (mod\ } {112})$ is representable by both or none of $x^{2}+14y^{2}$ and $x^{2}+448y^{2}$, whereas a prime $p\equiv 9,25,57\mathrm {\ (mod\ } {112})$ is representable by exactly one of these forms.
• A prime $p\equiv 1,169\mathrm {\ (mod\ } {240})$ is representable by both or none of $x^{2}+150y^{2}$ and $x^{2}+960y^{2}$, whereas a prime $p\equiv 49,121\mathrm {\ (mod\ } {240})$ is representable by exactly one of these forms.

It is conjectured that there are no other similar results involving definite forms.