Fermat's Last Theorem!
Even if you're not into math, you've probably heard of this one. Back in 1637, Pierre de Fermat conjectured that no three positive integers a, b, and c could satisfy the equation a^n + b^n = c^n for any integer value of n greater than two. He scribbled this conjecture in the margin of a copy of Arithmetica. Fermat went on to claim he had a proof for this, but it was too long to fit in the margin. By the time of his death, he had not produced this proof.
A few mathematicians tackled this problem over the years. Okay, more than a few. More like nearly every mathematician for the next 358 years. Finally, in 1995, Andrew John Wiles managed to prove it... his proof was over 100 pages long, took seven years of research, and made use of techniques that did not exist in Fermat's time. Wiles was knighted for his contribution to mathematics.
I remember my first encounter with the Last Theorem. Back in high school, in the first linear algebra class I'd ever taken, on the first day of class, we were asked disprove the theorem as homework, the equation scrawled on the whiteboard in the last few moments of class as if it was no big deal. Now, back in those days my Google-fu was weak and my work ethic was strong, so I went at it with a gusto. And I arrived at class the next day feeling utterly despondent, having found exactly zero solutions and having no idea why. Oh math... I think that's the day I fell in love with you.
I feel like I should talk about Fourier analysis and how it can do anything if you're clever enough, or Fibonacci numbers and how they're already doing everything whether you're clever or not... but instead, I'll just ask this: do you think Fermat had a solution? Or was he just another mathematical rebel without a cause (more on that tomorrow)?