I once read somewhere, (can't find link) that to prove Fermat's Last Theorem, assuming it has been proven for $n = 3, 4$, it suffices to prove it for every prime $n \ge 5$. I have no idea why this is true. Can somebody explain?

  • $\begingroup$ What is FLT???? $\endgroup$ – JPi Mar 20 '14 at 22:06
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    $\begingroup$ $a^{pq}+b^{pq}=(a^q)^p+(b^q)^p$ $\endgroup$ – Ian Mateus Mar 20 '14 at 22:07
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    $\begingroup$ @JPi: Fermat's Last Theorem. $\endgroup$ – Asaf Karagila Mar 20 '14 at 22:07

Suppose $n$ is not prime, then it can be written as $pq$ for some $p,q\in\mathbb{N}$, with $q$ prime, and $a^n$ can be written as $(a^p)^q$ so if there is a solution for $n$ there also is one for $q$ (which is smaller than $n$).

  • $\begingroup$ The OP is asking why it suffices to prove it for primes greater than $4$. $\endgroup$ – user71641 Mar 20 '14 at 22:06
  • $\begingroup$ Did you even read the question? $\endgroup$ – user85798 Mar 20 '14 at 22:07
  • $\begingroup$ @ABC: OJB was right, I corrected my answer after because I had initially misread the question $\endgroup$ – Alessandro Codenotti Mar 20 '14 at 22:15
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    $\begingroup$ $a^{p^q} \neq \left(a^p\right)^q$. $\endgroup$ – Eric Towers Mar 20 '14 at 22:30
  • $\begingroup$ @Eric: Thank you for the correction, I updated the answer $\endgroup$ – Alessandro Codenotti Mar 20 '14 at 22:37

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