Re: [PrimeNumbers] Euler 6n+1 and Fermat 4n+1 theorems
- In an email dated 29/5/2005 9:44:59 am GMT Daylight time, najiba amimar <najibaamimar@...> writes:
>Every prime number of the form 6n+1This is partly because the proofs are not that straightforward!
>can be written as x**2 + 3 * y**2 (Euler)
>every prime number of the form 4n+1
>can be written as x**2 + y**2 (Fermat)
>These two theorems have been demonstated
>more than two century ago, but the proofs
>seems tremendously difficult to find on
As so often, Hardy & Wright is a good source.
They give four distinct proofs of a more general theorem than the one you refer to:-
Theorem 366. A number n is the sum of two square if and only if all prime factors of n of the form 4m+3 have even exponents n the standard form of n.
>>Perhaps the simplest method is by using the properties of the Gaussian primes, viz. those of Z[i].
The analogous theorem for numbers of the form 3m+1 resp. 3m+2 can be found and proved by investigating the Eisenstein primes, viz. those of the ring Z[w], where w = -1/2+sqrt(-3)/2.