3
$\begingroup$

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?

$\endgroup$
  • 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
6
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.