Graduate Prime Number Powers of Integers and Fermat's Last Theorem

[e2m2a]

TL;DR

Need understanding of an integer to a perfect power raised to the power of a prime number as it relates to Fermat's Last Theorem.

From my research I have found that since Fermat proved his last theorem for the n=4 case, one only needs to prove his theorem for the case where n=odd prime where c^n = a^n + b^n. But I am not clear on some points relating to this. For example, what if we have the term (c^x)^p, where c is an integer, x = even integer, and p = odd prime. Then we can express this term as c^(xp) and we would have c^(xp)=a^n +b^n. Clearly, xp is no longer an odd prime. So, does this mean to prove Fermat's Last Theorem for the case where n=odd prime, then neither of the bases c,a,b themselves can be an integer that is raised to an even numbered power?

 

[mfb]

If $c^{xp}=a^{xp} +b^{xp}$ is a counterexample for the exponent n=xp then $(c^x)^p = (b^x)^p + (a^x)^p$ is a counterexample with exponent p. If you can show that there is no counterexample for p then there can't be a counterexample for np for any positive n. It doesn't matter what a,b,c are.

Every integer larger than 2 is a multiple of an odd prime or a multiple of 4, so we only need to show that there are no counterexamples for odd primes and for the number 4.