I'm looking at the wikipedia page on Fermat's Last Theorem
In the statement it requires $a,b,c$ to be positive integers. Is that correct? I always took it to be no solutions in non-zero integers. But this wiki page makes a big deal out of the bases being positive. Has some counter-example turned up using negative integers that I'm missing? Otherwise, I think we should fix the wiki page.
Best Answer
The formulations are equivalent. This is clear when $n$ is even (because then $x^n=(-x)^n$), so assume $n$ is odd. I will prove that if FLT has no positive solutions, it will have no non-zero solutions. We have few cases:
$a,b,c>0$ - we know this doesn't have solutions.
$a,b,c<0$ - if these were a solution, we would have $a^n+b^n=c^n$ and then by multiplying by $(-1)^n$ we would have $(-a)^n+(-b)^n=(-c)^n$ with $-a,-b,-c>0$.
$a,b<0, c>0$ - this is impossible, as then $a^n+b^n<0<c^n$
$a,b>0, c<0$ - same as above.
$a>0,b<0,c>0$ - if we have $a^n+b^n=c^n$, then $c^n+(-b)^n=a^n$.
Rest of the cases goes similarly, which I will leave for you.