The theorem is stated here: http://en.wikipedia.org/wiki/Fermat's_Last_Theorem

Fermat's Last Theorem...states that no three positive integers $a$, $b$, and $c$ can satisfy the equation $a^n$ + $b^n$ = $c^n$ for any integer value of n greater than two.

What is known about solutions for any non-zero integers $a$, $b$, and $c$, and any integer $n > 2$? (not just a restriction to only positive integers). Are there any solutions? If so, are there only finitely many? Infinitely many?

  • 2
    $\begingroup$ It is easy to show that solutions in negative integers correspond to solutions in positive integers $\endgroup$ – Asier Calbet Aug 6 '14 at 14:33
  • $\begingroup$ @Assaultous2 What is your reasoning? $\endgroup$ – Ryan Aug 6 '14 at 14:35
  • $\begingroup$ @Assaultous2 I've deleted my answer which was essentially the same as your comment - you should put yours up as an answer. $\endgroup$ – Mark Bennet Aug 6 '14 at 14:40

For even $n$, changing the sign of $a,b,c$ does not change the equality in any way. For odd $n$, after the signs of each term has been calculated, all terms can be moved to another side if necessary to have an equation with all $a,b,c$ positive. Either they are all on one side, which is only possible if $a=b=c=0$, or there are 2 on one side and 1 on the other, which corresponds exactly to one of the positive solutions.


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