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Extending the 10 squares conjecture

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  • Jon Perry
    Into various permutations of the original: http://www.users.globalnet.co.uk/~perry/maths/extendingtensquares/extendingt ensquares.htm including quadruples such
    Message 1 of 5 , Aug 5, 2002
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      Into various permutations of the original:

      http://www.users.globalnet.co.uk/~perry/maths/extendingtensquares/extendingt
      ensquares.htm

      including quadruples such that every combo of 3 is a square, e.g.:

      1,5,7,24
      1,8,45,91

      and the new conjecture that:

      For n>2, the simultaneous equations:

      ab+1=x^n
      ac+1=y^n
      bc+1=z^n

      have no solutions.

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry/maths
      BrainBench MVP for HTML and JavaScript
      http://www.brainbench.com
    • jbrennen
      ... Oh, really. I found a solution. :-) (a,b,c,x,y,z,n) == (2,171,25326,7,37,163,3) To effectively search this, don t vary (a,b,c). Instead choose n, then
      Message 2 of 5 , Aug 5, 2002
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        --- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
        > For n>2, the simultaneous equations:
        >
        > ab+1=x^n
        > ac+1=y^n
        > bc+1=z^n
        >
        > have no solutions.

        Oh, really. I found a solution. :-)

        (a,b,c,x,y,z,n) == (2,171,25326,7,37,163,3)


        To effectively search this, don't vary (a,b,c). Instead choose n,
        then vary (y,x,a). y goes from 3 to LIMIT; x goes from 2
        to y-1; a iterates over the divisors of gcd(x^n-1,y^n-1).

        Pari/GP:

        n=3;for(y=3,10000,yt=y^n-1;for(x=2,y-1,xt=x^n-1;g=gcd(xt,yt);
        if(g>1,p=xt*yt;fordiv(g,a,if(a>1,w=p/(a^2)+1;
        z=round(w^(1/n));if(w==z^n,b=(x^n-1)/a;c=(y^n-1)/a;
        print("(",a,",",b,",",c,",",x,",",y,",",z,",",n,")")))))))

        The solution above takes less than one second to find.
      • Jon Perry
        (a,b,c,x,y,z,n) == (2,171,25326,7,37,163,3) Nice work. It s still unsolved for n 3 though. See:
        Message 3 of 5 , Aug 5, 2002
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          (a,b,c,x,y,z,n) == (2,171,25326,7,37,163,3)

          Nice work. It's still unsolved for n>3 though.

          See:

          http://www.users.globalnet.co.uk/~perry/maths/extendingtensquares/extendingt
          ensquares.htm

          Jon Perry
          perry@...
          http://www.users.globalnet.co.uk/~perry/maths
          BrainBench MVP for HTML and JavaScript
          http://www.brainbench.com
        • Phil Carmody
          ... Smart method. n=4 (1352,9539880,9768370,337,339,3107,4) Phil ===== -- The good Christian should beware of mathematicians, and all those who make empty
          Message 4 of 5 , Aug 5, 2002
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            --- jbrennen <jack@...> wrote:
            > --- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
            > > For n>2, the simultaneous equations:
            > >
            > > ab+1=x^n
            > > ac+1=y^n
            > > bc+1=z^n
            > >
            > > have no solutions.
            >
            > Oh, really. I found a solution. :-)
            >
            > (a,b,c,x,y,z,n) == (2,171,25326,7,37,163,3)
            >
            >
            > To effectively search this, don't vary (a,b,c). Instead choose n,
            > then vary (y,x,a). y goes from 3 to LIMIT; x goes from 2
            > to y-1; a iterates over the divisors of gcd(x^n-1,y^n-1).
            >
            > Pari/GP:
            >
            > n=3;for(y=3,10000,yt=y^n-1;for(x=2,y-1,xt=x^n-1;g=gcd(xt,yt);
            > if(g>1,p=xt*yt;fordiv(g,a,if(a>1,w=p/(a^2)+1;
            > z=round(w^(1/n));if(w==z^n,b=(x^n-1)/a;c=(y^n-1)/a;
            > print("(",a,",",b,",",c,",",x,",",y,",",z,",",n,")")))))))
            >
            > The solution above takes less than one second to find.

            Smart method.
            n=4 (1352,9539880,9768370,337,339,3107,4)

            Phil


            =====
            --
            The good Christian should beware of mathematicians, and all those who make
            empty prophecies. The danger already exists that the mathematicians have
            made a covenant with the devil to darken the spirit and to confine man in
            the bonds of Hell. -- Common mistranslation of St. Augustine (354-430)

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          • Jon Perry
            ... I agree, see same page for quote. P.S. I hope you are all checking the case for 4 variables (nay 5...).... Jon Perry perry@globalnet.co.uk
            Message 5 of 5 , Aug 5, 2002
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              >Smart method.
              >n=4 (1352,9539880,9768370,337,339,3107,4)

              I agree, see same page for quote.

              P.S. I hope you are all checking the case for 4 variables (nay 5...)....

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