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Re: Complex a*x^n+b*y^n puzzle

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  • djbroadhurst
    ... These two question appear to be identical; the second merely changes the word order and the labelling in the first, without altering the mathematics in any
    Message 1 of 37 , Dec 4, 2009
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      --- In primenumbers@yahoogroups.com,
      "Robdine" <robdine@...> asked:

      > Is a gaussian number a+bi prime if a^2+b^2 is a (integer)prime
      > and if c^2+d^2 is prime then is also c+di a gaussian prime?

      These two question appear to be identical;
      the second merely changes the word order
      and the labelling in the first, without
      altering the mathematics in any respect.

      In each case, the answer is yes.

      Moreover, if a and b are non-zero rational integers, then
      a + I*b is a Gaussian prime if and *only* if
      a^2 + b^2 is a rational prime.

      Note, however, that 3 + I*0 is a Gaussian prime,
      but 3^2 + 0^2 is not a rational prime.

      To determine whether z is a Gaussian prime
      we may use the following procedure:

      {isgp(z)=local(a,b,c,t=0);
      a=abs(real(z));b=abs(imag(z));
      if(type(a)=="t_INT"&&type(b)=="t_INT",c=max(a,b);
      if(a*b,t=isprime(a^2+b^2),t=(c%4==3)&&isprime(c)));t;}

      for(a=1,7,for(b=0,a,z=a+I*b;if(isgp(z),print1(z", "))));

      1 + I, 2 + I, 3, 3 + 2*I, 4 + I, 5 + 2*I, 5 + 4*I, 6 + I, 6 + 5*I, 7, 7 + 2*I,

      David
    • djbroadhurst
      ... There is only rational prime of the form a^2 + b^2 that yields precisely 4 distinct Gaussian primes, namely 2 = 1^2 + 1^2. If a^2 + b^2 is an odd rational
      Message 37 of 37 , Dec 5, 2009
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        --- In primenumbers@yahoogroups.com,
        "Robdine" <robdine@...> wrote:

        > any rational prime that can be represented by the sum
        > of 2 squares (a^2+b^2) will define 4 gaussian primes

        There is only rational prime of the form a^2 + b^2 that yields
        precisely 4 distinct Gaussian primes, namely 2 = 1^2 + 1^2.

        If a^2 + b^2 is an odd rational prime, we have
        8 [sic] asociates of the Gaussian prime z = a + I*b,
        since we may mulitply it and its conjugate z = a - I*b
        by the 4 units I, -I, -1, 1, obtaining 8 distinct
        Gaussian integers that are prime.

        David
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