Fermat's Last Theorem
Fermat's Last Theorem was proposed by Pierre de Fermat in the in the margin of his copy of the book Arithmetica. The note in the margin (when translated) read: "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
Many mathematicians today doubt that Fermat actually had a proof for this theorem. If he did have one, he never published it, though he did publish a proof for the case . It seems unlikely that he would have circulated a proof for the special case when he had a general solution. Some think that Fermat's proof was flawed, and that he saw the flaw after a time.
Some mathematicians have suggested that Fermat had a proof that relied on unique factorization in rings of the form . Unfortunately, this is not often the case. In fact, it has now been known for some time how to solve the problem when this is the case. See this video link for detailed explanation of the proof and the concept of unique factorization: https://youtu.be/jfDbnz-Bp_g
Despite Fermat's claim that a simple proof existed, the theorem wasn't proven until Andrew Wiles did so in . Wiles's proof was the culmination of decades of work in number theory. Interestingly enough, Wiles's proof was much more modern than anything Fermat could have produced himself. It exploited connections between modular forms and elliptic curves.
In some sense, Fermat's last theorem is a dead end: it has led to few new mathematical consequences. However, the search for the proof of the theorem generated whole new areas of mathematics. In this sense, it was a good, productive problem.
The ABC Conjecture is a far-reaching conjecture that implies Fermat's Last Theorem for . It is one of the most famous still-open problems in number theory, and despite various attempts to prove the conjecture, as of none of these attempted proofs have been accepted and the ABC conjecture is still considered unproven.