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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 ?

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up vote 6 down vote accepted

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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The two problems are equivalent: since the polynomial equation is homogeneous (i.e. all summands have the same degree), you can clear denominators.

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Fermat's last theorem is that there are no solutions to $x^n+y^n=z^n$ for $x, y,z$ positive integers and $n\ge 3$.

Let's set $\displaystyle a:=\frac xz,\ b:=\frac yz\ $ then an equivalent formulation is that $a^n+b^n=1$ admits no non trivial rational solutions (i.e. other than $(1,0),\;(0,1),\;(-1,0),\;(0,-1)$).

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