# Fermat's Last Theorem - A query

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 ?

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.

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

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 $(1,0),\;(0,1),\;(-1,0),\;(0,-1)$).