In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

quadratic forms In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

. It was proved in 2003 by

Irving Kaplansky Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...

Statement of the theorem

Kaplansky's theorem states that a prime ''p'' congruent to 1 modulo 16 is representable by both or none of ''x''² + 32''y''² and ''x''² + 64''y''², 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.

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 + 64''y''², and that −4 is an 8th power modulo ''p'' if and only if ''p'' is representable by ''x''² + 32''y''².

Examples

*The prime ''p'' = 17 is congruent to 1 modulo 16 and is representable by neither ''x''² + 32''y''² nor ''x''² + 64''y''². *The prime ''p'' = 113 is congruent to 1 modulo 16 and is representable by both ''x''² + 32''y''² and ''x''²+64''y''² (since 113 = 9² + 32×1² and 113 = 7² + 64×1²). *The prime ''p'' = 41 is congruent to 9 modulo 16 and is representable by ''x''² + 32''y''² (since 41 = 3² + 32×1²), but not by ''x''² + 64''y''². *The prime ''p'' = 73 is congruent to 9 modulo 16 and is representable by ''x''² + 64''y''² (since 73 = 3² + 64×1²), but not by ''x''² + 32''y''².

Similar results

Five results similar to Kaplansky's theorem are known:. *A prime ''p'' congruent to 1 modulo 20 is representable by both or none of ''x''² + 20''y''² and ''x''² + 100''y''², 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 ''x''² + ''xy'' + 10''y''² and ''x''² + ''xy'' + 127''y''², 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 ''x''² + ''xy'' + 14''y''² and ''x''² + ''xy'' + 69''y''², 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 ''x''² + 14''y''² and ''x''² + 448''y''², 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 ''x''² + 150''y''² and ''x''² + 960''y''², 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

{{reflist Theorems in number theory Quadratic forms