representing primes as
Theorem 1 (Fermat).
An odd prime can be written with if and only if .
: This direction is obvious. Since is odd, exactly one of is odd. If (say) is odd and is even, then and .
: Since , by Euler’s Criterion we have that where is the Legendre symbol. Choose such that . Working in we have . Then , but does not divide either factor since . Hence is not prime. Since is a UFD, it follows that is not irreducible either, so we can write , where neither factor is a unit (i.e. neither factor has norm ). Taking norms, we get
Since neither factor has norm , we must have , so is the sum of two squares. ∎
There are more elementary proofs of , but one can try to generalize the given proof for arbitrary . When can be written as ? By analogy with the proof for , suppose we find such that (i.e. that ). Then in , it follows that , so again is not prime since it does not divide either factor. If is a UFD, then is not irreducible either. We can then write as before and, taking norms, we get the same result: .
This argument relies on two things: first, that is a square (i.e. that ); second, that is a UFD. It is known that the only imaginary quadratic rings that are UFDs are those for , and the only in that set for which is the ring of integers are .
So for , and an odd prime, if and only if , while for the other (), the ring of integers of is not , so is not integrally closed and thus is not a UFD and hence this proof will not work for those values of .
The cases and can be dealt with by the following relatively simple argument (which, as you can see, does not generalize further): Corollary 6 in the article on representation of integers by equivalent integral binary quadratic forms states that if is an odd prime not dividing , then if and only if is represented by a primitive form (http://planetmath.org/IntegralBinaryQuadraticForms) of discriminant (http://planetmath.org/RepresentationOfIntegersByEquivalentIntegralBinaryQuadraticForms) . So if there is only one reduced form (http://planetmath.org/ReducedIntegralBinaryQuadraticForms) with that (which must perforce be the ), then we are done. But (see this article (http://planetmath.org/ThereIsAUniqueReducedFormOfDiscriminant4nOnlyForN12347)). and were dealt with above. is the form , and if an odd prime can be written , then clearly we have also ; conversely, if , then either or is even, so that also . Thus has the same set of solutions as . But we do get two new cases, and , for which have shown that is representable as if and only if , i.e. is a square . We have thus proven
If , , , , or , then an odd prime can be written as with , if and only if
i.e. if and only if is a square .
|Title||representing primes as|
|Date of creation||2013-03-22 15:49:28|
|Last modified on||2013-03-22 15:49:28|
|Last modified by||rm50 (10146)|