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4272Re: [PrimeNumbers] Pythagoras and factoring

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  • Aleksey D. Tetyorko
    Dec 3, 2001
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      Hi, David & All!

      You method (IMHO) based upon the following:
      Let A be the number to factor. Using common solution to Pythagoras, we can
      write:

      A=U^2-V^2
      B=2*U*V
      C=U^2+V^2

      so that A^2+B^2=C^2.

      Then B+C=(U+V)^2, B-C=(U-V)^2 as in your examples. But it is *not* common
      solution!
      For instance, the least solution to A^2+B^2=C^2 with A=123 is

      123^2+164^2=205^2

      Both your C-program and my pascal one find this solution. But,alas

      205-164=41 is *not* square, neither is
      205+164=369=3^2*41.

      All this nightmare is due to fact that common solution to Pythagoras is

      A=K*(U^2-V^2)
      B=2*K*U*V
      C=K*(U^2+V^2)

      and, for instance, B+C=K*(U+V)^2. Only GCD can help, or another solution to
      Pythagoras.

      Aleksey