Problem Statement:
In Fermat's Last Theorem $$x^n + y^n = z^n$$ $x,y,z$ are considered integers. But upon closer inspection it is seen that it is also true for any rational numbers $x,y,z$. And that FLT is not applicable only when $x,y,z$ are irrational.
Query : Why is it that then it is always and only mentioned that Fermat's theorem is true when $x,y,z$ are integers and not rational numbers ? Is my perception correct? Can this be proven or disproved ?
Answer
As MTurgeon points, the two problems are equivalent.
More exactly, for some $n$, the equation $x^n+y^n=z^n$ has non trivial integer solutions if and only if $x^n+y^n=z^n$ has non trivial rational solutions.
Anyhow, many of the techniques used to attempt a proof, both in general and in the particular cases, work for the integer version. For example, the case $n=3$ relays on the fact that $\mathbb{Z}(\omega)$ is an UFD, the case $n=4$ is based on the fact that one gets a contradiction by building a smaller positive solution.
Since the two problems are equivalent, and in the study the integer version is easier to approach, it is typically posted as an equation over the integers.
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