One interesting historical note on Fermat's Last Theorem.

In 1637, in Fermat's book Arithmetica, he discussed integer solutions to $$$x^2 + y^2 = z^2$$$, which I think at the time was well known. At least, it's fairly standard intro level number theory, nowadays. Now, the more interesting generalization is solutions to the equation $$$x^n + y^n = z^n$$$ in $$$\mathbb{Z}$$$. And it certainly appears that this has no solutions when $$$n > 2$$$, so Fermat made the remark that he speculated it had no solutions. He said the general solutions was too long to fit into the margins of the textbook. But this turns out to be a really hard problem, and the general consensus is that Fermat was not able to solve it and bluffed (i.e. the equivalent of saying "proof is left as an exercise to the reader" when you don't know how to solve the problem). Why is this the consensus?

It turns out even case $$$n =3, 4$$$ is pretty hard. Euler solved the case of $$$3$$$ in the late $$$1700s$$$, Legendre solved case of $$$5$$$ in 1825, and Kummer later proved the case of $$$4$$$ (Germain proved a case of the case of $$$4$$$ earlier, but with the restriction that $$$a,b,c$$$ are prime). Point is: the problem is hard. Even individual cases of $$$n$$$ took centuries.

And eventually, in 1995, Andrew Wiles proved the statement. The proof exceed 100 pages, and used lots of theory Fermat was not aware of at the time (stuff like modular elliptic curves, nontrivial ring/group theory, etc).

So yeah, TL;DR Fermat basically bluffed and pretended to be able to solve it.