## Extending the 10 squares conjecture

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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
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
• ... 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.
• (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
• ... 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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• ... 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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