Now that Fermat’s theorem has finally been proven. What is the next great theorem that needs to be proved?
For those of you that don’t know Fermat around 1637, the French jurist Pierre de Fermat scribbled in the margin of his copy of Diophantus’ Arithmetica what came to be known as Fermat’s Last Theorem, the most famous question in mathematical history. Stating that it is impossible to split a cube into two cubes, or a fourth power into two fourth powers, or any higher power into two like powers, but not leaving behind the marvelous proof he claimed to have had, Fermat prompted three and a half centuries of mathematical enquiry which culminated only recently with the proof of Fermat’s Last Theorem by Andrew Wiles.
I don’t know what mathematicians consider the next great question, but could it possibly be the Goldbach conjecture that every even number greater than two is the sum of two (not necessarily distinct) primes?
If it hasn’t been shown that [sym]p[/sym] is normal, then that’s on the list too. It doesn’t quite have the status of the millennium problems, but it would still be nice to know.
As far as practical matters go, it’s hard to imagine something more important than P vs. NP.
I second the Goldbach conjecture as the Big Open Problem in Mathematics. But my reasons may not be the same as others.
It is easy to state. Try explaining the Extended Riemann Hypothesis in one sentence. Most people are “aware of” terms like “even” and “prime” (but probably fuzzy about the details). So they can appreciate to some extent how simple it is to phrase while apparently hard to solve. Much like former favorites such as Four Color Theorem or FLT.
It is finitely refutable. I.e., a single, finite, counter-example disproves the conjecture. In CS, I can point out to students how easy it is to write a program to search for a counterexample. It halts iff the conjecture is false. So determining whether a program, even without inputs, halts can be extremely difficult. Makes getting them to accept halting problem unsolvability much easier.
As an old time CS hand, I am constantly surprised to see “P=NP?” pop up on such lists. It is a major open question in Computer Science and not Mathematics. It is as jarring as if you saw questions about room-temperature superconductors or life on mars on a Math list. It just doesn’t belong.
where the sum is from i=1 to infinity (or maybe zero to infinity, I can’t remember offhand). Extend this function to all of the complex numbers. The question is, when is this function equal to zero? The conjecture is that every point where this is equal to zero has real part 1/2 (or maybe imaginary part 1/2, I get that backwards sometimes).
This function is fundamentally linked with the prime numbers. If it it is proven true, a whole slew of results in number theory will also be proven as a result. If it is proven false, a whole slew of results in number theory will still be proven true. Either way, the gained knowledge in number theory (and probably other fields) will be huge; we’ll learn a lot more about the primes from this one result.
If Mr. Wiles proof is the one that PBS had a documentary on where they fellow took up to 300 pages to prove Fermat’s Last Therom, then the next cute question was what was Fermat thinking? Certainly not a 300 page answer, and was he right?
The most accessable replacement is clearly Goldbach’s conjecture, but the Riemann hypothesis is surely the bigger prize to bag.
My impression is that the concensus amongst historians of mathematics is that Fermat had a habitual variation on proof by induction whereby he reduced theorems to problems of a lesser degree and then proved them for one of the “simple” cases. It’s relatively easy to find proofs of Fermat’s Theorem for low degrees (such as cubic powers) and the presumption is that he’d one, or several, such instances nailed and then failed to realise that the induction to all powers didn’t quite work. What he’d done was nontrivial for the period, but in trying to push it beyond the first few powers he was just wrong.
Thanks Bonzer. I’m not a math person, but I loved how they put together that documentary and the fellow’s passion for it really came through. He apparently solved some other sticky problems in the process.
If I understand you correctly, Fermat wasn’t doing what modern math would call a proof, but rather an example.
Fermat’s proofs in number theory were sound enough. His pet method of proof was a form of mathematical induction invented by him and called infinite descent. Basically, his method aims to show that a solution to a problem implies another solution with smaller values. But a sequence of decreasing postive integers cannot continue indefinitely.
Fermat himself used the method to prove x[sup]4[/sup] + y[sup]4[/sup] = z[sup]4[/sup] to be impossible in positive integers. He assumed the existence of a solution (x,y,z) and deduced from this another equation of the same form with solution of smaller values (X,Y,Z). Clearly this logical process can be repeated; equally clearly there are only a finite number of positive integers less than any given positive integer. So the assumption of the truth of the theorem is false.
I personally don’t belive Fermat had a proof of his famous theorem. I think he knew he had no proof too. Otherwise he would have challenged some other mathematician to prove it, a common practice at the time. I think he found the flaw in his proof and never bothered to erase his comment from his copy of Diophantus.
Unless I’m mistaken, that derivation has been shown not to exist, so it shouldn’t really count. On the other hand, I suppose it would be pretty impressive to derive something that’s untrue.
was missing from AHunter3’s post.