Prime Number Powers of Integers and Fermat's Last Theorem

In summary, it has been proven that for Fermat's Last Theorem, it is only necessary to prove the case where n is an odd prime, as the case for n=4 has already been proven. However, there are some questions regarding this, such as what happens if the bases c,a,b are integers raised to even powers. It has been shown that if there is a counterexample for n=xp, then there is also a counterexample for p. Therefore, in order to prove Fermat's Last Theorem for the case of odd primes, it is necessary to show that there are no counterexamples for p, which covers all odd primes. Additionally, it is also necessary to show that there are no counterexamples for
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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?
 
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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.
 
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