Why was proving Fermat’s Last Theorem such a big deal?

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Why was proving Fermat’s Last Theorem such a big deal?

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Because it was a statement generally thought to be true, but it hadn’t been proven for so long. Just like movies or games that people get hyped for years over, when it was finally solved there was a huge release of excitement just because… it was FINALLY solved.

Also, it took so long to solve because mathematics lacked the tools to do so, the techniques used to prove the theorem (which I don’t understand well enough to detail) involved a certain amount of combining existing techniques and inventing new ones. I know the result was finally proven using elliptic curves, and that since then the study of elliptic curves has advanced other research. So it was a noteworthy achievement that others reference to make new discoveries

1. Simple mathematical statements that are hard to solve are compelling in general, especially to mathematicians. It *feels* like simple statements should be easy to solve, so when we come across ones that aren’t, especially ones in which a solution feels obvious, they draw a lot of attention.
2. To add to the mystery is the fact that Fermat claimed to have a simple solution but it was never found and made #1 all the more frustrating.
3. The actual solution by Wiles was far from simple. It was rather complex delving into mathematical areas that no one would have every thought had anything to do with the problem or had anything to do with each other.

Here’s the claim:

> It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.[30][31]
https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem

I.e. you can’t have a^n + b^n = c^(n), where a, b, c, and n are integers, with n>2. The claim seems reasonable enough, and was proved for various values of n, and was finally proven in general in the 20th century. But *nobody* has come up with that simple, elegant proof that Fermat claimed to have. So … did he see something that everyone else has missed, or did he just make a mistake about that.