- 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 - --- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
> For n>2, the simultaneous equations:

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

>

> ab+1=x^n

> ac+1=y^n

> bc+1=z^n

>

> have no solutions.

(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:

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 - --- jbrennen <jack@...> wrote:
> --- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:

Smart method.

> > 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.

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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Yahoo! Health - Feel better, live better

http://health.yahoo.com >Smart method.

I agree, see same page for quote.

>n=4 (1352,9539880,9768370,337,339,3107,4)

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