Did great Mathematicians of the Past Leave Unsolved Problems Behind?

[ultrafilter]

@qts

IIRC Simon Singh’s book says that Fermat’s likely solution was the one worked on by Cauchy et al but disproved by Kummel - it supposed unique factorisation. But did Complex Numbers antedate Fermat?

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

[Cabbage]

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

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis’ De Algebra tractatus.

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.

[ultrafilter]

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.

[Mangetout]

@The_Llama

OK, I’m obviously wrong. I had to have misunderstood what she said. Perhapse some part of the final outcome was determined by proving something else false. All of the interviews I’ve been checking with Wiles say it’s proved, but nothing about proving anything false. Sorry, I honestly held this to be what I knew.

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.

[bonzer]

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.

[bonzer]

@ralph124c

So are modern math wizards still stumped by problems from 100-200 years ago?

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 …

[ITR_champion]

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

@archimedes

If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.
But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.

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.

[Mathochist]

@ITR champion

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

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.

[Chronos]

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?