Fermat, in mathematical circles, was a bit of an jerk. He was a member of a circle of thinkers (centered around Marin Mersenne) who exchanged letters on mathematics and other topics which, along with a few others, would form today’s modern mathematics and science.
Did I mention he was an jerk? Yes, about that. When you’re talking about mathematics, it’s not cool just to spout out random facts. You have to prove them. The beauty isn’t just some cool fact about numbers, but why numbers behave this way. Fermat, however, was prone to simply spouting off about what he was thinking about. Fermat didn’t prove his thoughts, he just skipped to stating them as facts, so he gave off the impression of being a know-it-all asshole. His approach was smug, and condescending, saying “I know something, but I’m not going to tell you, because it should be obvious, and you’d just agree with me if you were smart.” I recall that Mersenne actually threatened to cut him off from the group unless he shaped up. I get the picture that he was smart, and said amazing things that were correct, so people put up with his crap anyways.
A diversion, with a bit of math. We know that squares of whole numbers can add up to other squares. For example, the areas of squares of 3 and 4 (with areas of 9 and 16) add up to 25, which is the area of a square of 5. Fermat is playing with concepts that flow from this, and is thinking about cubes, and beyond. He has a thought, and writes in the margin of his book…”I have a marvelous proof” that sums of cubes (and higher powers of numbers) don’t add up evenly. But he’s a lazy asshole, and doesn’t write it out, using the crappy excuse that the margin of the book is too small. This is like borrowing $5 for a burger because “All I have is a $100 bill”. Scumbag Steve Fermat.
So a guy who’s known for unproven crazy accusations and know-it-all statements out of left field, but is frustratingly correct, makes another crazy statement, hiding behind a lame excuse, after his death. On one hand, it makes us want to find his answer. It’s caused mathematicians incredible efforts and countless hours over the years. It started with a curiosity, and an attempt to confirm or deny a bloviating jerk. And given the length of time it took all of us, Fermat’s original ‘proof’ was probably incorrect.
But, curse his rotten soul, it turns out the bastard was still right. Again. What a schmuck. His ghost is probably sitting back, drinking over-priced scotch, right now. And he’s thinking to himself “You imbeciles. My horse could have figured that out!”
In addition to the fact that dozens have tried in vain to prove it since the 1630s (until Wiles proved it in 1994), it shows that the Pythagorean Theorem has no generalizations for higher dimensions (i.e. higher n values). The significance is not the individual result confirming the truth of this particular theorem, but the path that Wiles took to get his result. His work in pursuit of the proof led to significant developments in algebraic geometry and number theory, which are very important fields.