Here, I will try to prove a special case of Fermat's Last Theorem, namely when $a=b$ in this definition:
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than two.
Here's the proof:
Let's suppose the equation $x^n+x^n=y^n$ have one or more solutions where $n$ is an integer grater than $2$ and $x$ and $y$ are also integers. This means that $2x^n=y^n$, thus $\sqrt[n]{2}x=y$. But this is a contradiction since $\sqrt[n]{2}$ is irrational for every $n$ in $\Bbb N$, which implies that $y$ is irrational contradicting our second assumption. Therefore, the equation $x^n+x^n=y^n$ has no solution in $\Bbb N$.
Do you think it is right?
Thank you.