Did great Mathematicians of the Past Leave Unsolved Problems Behind?

Yes, complex numbers were necessary to solve the cubic equation, which was done in the 1400s, IIRC.

ultrafilter, the first person I’m familiar with that really messed around with complex numbers was Cardano. He knew the cubic formula. Sometimes when he used it to solve a cubic equation with real solutions, intermediate steps in the formula involved imaginary numbers–yet still ultimately resulted in the correct solution. Cardano called these numbers “fictitious numbers”, yet still thought there might be something to them. He never quite figured it all out, however. This was back in the 16th century.

According to this link

Fermat was 17th century, which was apparently an intermediate time in the development of comlpex numbers. I don’t know whether he ever studied them or not.

It’s certainly possible that I’m mistaken–I’m no historian.

So if C wasn’t really popular until Gauss, Fermat would certainly never have run into it.

Perhaps you misunderstodd part of the Wiles Saga - he went public with what he believed to be a solution to Fermat’s Last Theorem, only to find that his work wasn’t actually watertight and it took him quite a lot more time and hard work to come back with a definitive proof.

Complex numbers are a distraction as far as Fermat and his “Last” Theorem are concerned.
The method alluded to by Chronos was the method of infinite descent, which in the cases of n=3 and 4 doesn’t require them. Furthermore, we’ve a pretty good idea that that’s what Fermat was thinking of in those two cases because he discusses them in these terms elsewhere in his writings. Without complex numbers coming into it. That this was Fermat’s line of reasoning is now accepted by historians of mathematics (possible excluding Singh) - see, for example, the discussion in The Mathematics Career of Pierre de Fermat (Princeton, 1973, 1994) by Michael Sean Mahoney.

Incidentally, my estimate would be that about 99%, and probably 99.9%, of all interesting mathematical problems over a century old have been solved. But it’s the remainder that are the good ones …

The great ancient Greek mathematician Archimedes left behind this little puzzler, which wasn’t solved until the late 20th century:

We now have a solution, though for reasons that will become obvious when you read this page, I am not going to post it here.

Your link itself puts a complete published solution technique in the late 19th century, and the technique itself surely wouldn’t have stumped a mathematician even before the problem came to modern attention. Just because a solution doesn’t give the number explicitly doesn’t mean it’s any less a solution.

But Archimedes seems to be implying that he knew the solution, which is obviously impossible. It seems reasonable to suppose that there’s an error somewhere, either in his math (he might have mistakenly thought that a number was square, or missed some of the arithmetic) or in the written form of the problem (like, maybe he meant to say “a half and two thirds”, not “a half and a third”). Has there been any speculation along these lines, of a simple error which would make the problem much more tractable?