--- 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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