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Re: 24n+1 is prime

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  • Adam
    As mentioned by others, you can prove x*y == 0 mod 3 but I am having a hard time connecting the relevance of this to the original question, uniquely. Adam
    Message 1 of 6 , Mar 28, 2003
      As mentioned by others, you can prove x*y == 0 mod 3 but I am having
      a hard time connecting the relevance of this to the original
      question, "uniquely."


      Adam

      --- In primenumbers@yahoogroups.com, "Jon Perry" <perry@g...> wrote:
      > Can someone help me prove this?
      >
      > If 24n+1 is prime, and as this is a prime 1mod4, then 24n+1 = x^2 +
      y^2
      > uniquely.
      >
      > {
      > for(n=1,1000,p=24*n+1;if (n%100==0,print(n));
      > if (isprime(p),sp=sqrt(p);
      > for (i=1,sp,for (j=i,sp,k=i*i+j*j;
      > if (k==p,ok=0;
      > if (i%3==0,ok=1);
      > if (j%3==0,ok=1);
      > if (ok==0,print(p": "i", "j" -> "ok)))))))
      > }
      >
      > This code confirms that for n to 1000, one of x or y is ALWAYS
      divisible by
      > 3.
      >
      > Jon Perry
      > perry@g...
      > http://www.users.globalnet.co.uk/~perry/maths/
      > http://www.users.globalnet.co.uk/~perry/DIVMenu/
      > BrainBench MVP for HTML and JavaScript
      > http://www.brainbench.com
    • Paul Jobling
      Fermat showed that every prime of the form 4n+1 can be expressed as the sum of two squares in exactly one way. As a prime of the form 24n+1 is 1 mod 3, x^2+y^2
      Message 2 of 6 , Mar 28, 2003
        Fermat showed that every prime of the form 4n+1 can be expressed as the sum of
        two squares in exactly one way.

        As a prime of the form 24n+1 is 1 mod 3, x^2+y^2 must be 1 mod 3, and that can
        happen if exactly one of them is divisible by 3.

        What is the problem here?


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      • Jon Perry
        Actually, Euler proved it, Fermat conjectured it: http://mathworld.wolfram.com/Fermat4nPlus1Theorem.html Jon Perry perry@globalnet.co.uk
        Message 3 of 6 , Mar 28, 2003
          Actually, Euler proved it, Fermat conjectured it:

          http://mathworld.wolfram.com/Fermat4nPlus1Theorem.html

          Jon Perry
          perry@...
          http://www.users.globalnet.co.uk/~perry/maths/
          http://www.users.globalnet.co.uk/~perry/DIVMenu/
          BrainBench MVP for HTML and JavaScript
          http://www.brainbench.com

          -----Original Message-----
          From: Paul Jobling [mailto:Paul.Jobling@...]
          Sent: 28 March 2003 15:09
          To: 'Adam'; primenumbers@yahoogroups.com
          Subject: RE: [PrimeNumbers] Re: 24n+1 is prime


          Fermat showed that every prime of the form 4n+1 can be expressed as the sum
          of
          two squares in exactly one way.

          As a prime of the form 24n+1 is 1 mod 3, x^2+y^2 must be 1 mod 3, and that
          can
          happen if exactly one of them is divisible by 3.

          What is the problem here?


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          Virus checked by MessageLabs Virus Control Centre.


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