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Re: [PrimeNumbers] Euler 6n+1 and Fermat 4n+1 theorems

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  • mikeoakes2@aol.com
    ... This is partly because the proofs are not that straightforward! As so often, Hardy & Wright is a good source. They give four distinct proofs of a more
    Message 1 of 2 , May 30, 2005
      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+1
      >can be written as x**2 + 3 * y**2 (Euler)
      >and
      >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
      >the web!

      This is partly because the proofs are not that straightforward!

      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.

      -Mike Oakes
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