Timeline for What is the oldest open problem in mathematics?
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
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| Oct 26 at 17:25 | comment | added | nealmcb | Veritasium has a wonderful video on exactly this! The Oldest Unsolved Problem in Math | |
| Dec 31, 2022 at 12:38 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Nov 26, 2022 at 17:25 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Fixed broken link
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| Nov 29, 2021 at 8:19 | comment | added | Jose Arnaldo Bebita | @NoahSnyder: In the Wikipedia section on odd perfect numbers, it is stated that "In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists." The cited reference is Dickson's History of the Theory of Numbers, Vol. I (1919), p. 6. | |
| Jun 7, 2010 at 14:23 | vote | accept | coudy | ||
| Jun 5, 2010 at 3:42 | comment | added | Nate Eldredge | @TonyK: I bet Nicomachus had a proof, probably a very remarkable one, but the margin of his papyrus... well, you know how it is. | |
| Jun 5, 2010 at 1:55 | comment | added | Noah Snyder | Ah, ok, you seem to be right that it's more accurate to say that the claim here is that Euclid's description gives all perfect numbers. Still, that's essentially the same open question. | |
| Jun 5, 2010 at 1:46 | comment | added | Victor Protsak | Right after "Some of the assertions are made in this quote about perfect numbers which follows the description of the algorithm". | |
| Jun 5, 2010 at 1:07 | comment | added | Noah Snyder | No, that's just the excerpt of point (4). They didn't excerpt the statements of the other points. | |
| Jun 5, 2010 at 0:52 | comment | added | Victor Protsak | Actually, the McTutor article does quote the passage from which they deduce "assertion (2)" that all perfect numbers are even. My take on it is that $\textit{particular numbers generated by Euclid's algorithm [described right before] are all even},$ which is true. The authors seem to say that there is nothing in that passage beyond illustrating Euclid's rule by the first four perfect numbers 6,28,496,8128. You can write to them and ask whether they had any other evidence. As it is, their quote does not support their attribution of assertion (2) to Nicomachus, in my opinion. | |
| Jun 5, 2010 at 0:18 | comment | added | Victor Protsak | At best, you can say that Nicomachus claimed that Euclid's rule gives all perfect numbers, which implies that (a) all even perfect numbers arise in this way (true, proved by Euler) and (b) there are no odd numbers (still unknown). What $\textit{I}$ am uncomfortable with is basing a categorization of X as a "problem" or "subject of inquiry" on $\textit{a posteriori}$ existence of a clean answer to it. | |
| Jun 5, 2010 at 0:09 | comment | added | Noah Snyder | So, you're claiming that the article I linked to is wrong, the excerpt you give from Dickson doesn't seem enough to draw that conclusion. I tried to find the original source on google books and it doesn't appear to be there. | |
| Jun 5, 2010 at 0:07 | comment | added | Victor Protsak | Yes, we are talking past other, I fear! When I say "P overlooked X", I truly mean "P didn't even consider the possibility of X", which seems to be the case here (only even numbers categorized as abundant, perfect, or efficient). Forget "credit", that may be a poor word choice. Suppose that P made a claim Z that, logically speaking, implies X. I agree that "X dates back to the work of P", but unless there is evidence that P considered X itself, and not Z, I wouldn't express the situation as "P falsely claimed to have solved X" or that "X appeared in the work of P". | |
| Jun 4, 2010 at 22:53 | comment | added | Noah Snyder | Also I think there's a difference between "crediting X with posing Y as an open problem (when they really said they'd answered Y)" and simply saying "Y is an open problem which dates back at least to the work of X (who falsely claimed to have solved it)." They're different in two ways, first "credit" is too positive a word for this situation, and second "posing X as an open problem" is only one of way that a problem can first appear in the literature. | |
| Jun 4, 2010 at 22:49 | comment | added | Noah Snyder | I feel like we're talking past each other. Nichomachus did explicitly ask these questions when he claimed to answer them! But on the specific point I don't think "find all even perfect numbers" is specific enough to be a "problem" (as opposed to a subject of inquiry), but I do think "does Euclid's formula give all even perfect numbers" is a specific problem. However, it's not clear from Euclid that he ever thought that formula gave all even perfect numbers, so I'm not comfortable crediting him with that problem. | |
| Jun 4, 2010 at 22:44 | comment | added | Victor Protsak | What I disagree with is that the problem of non-existence of odd perfect numbers was $\textit{raised}$ by Nicomachus: I don't see any evidence for that. For all we know he could have simply overlooked the possibility. There have been many wrong claims made in history of mathematics, including the perfect numbers problem, as enumerated by Dickson. Should we interpret each one of them as an implicit question? That would be rather unorthodox. | |
| Jun 4, 2010 at 22:43 | comment | added | Victor Protsak | An analogous situation with primes would be to give a $\textit{formula}$ for some primes, not just to consider primes. I don't see why "find all perfect numbers" would be illegitimate as a specific problem, $\textit{had it been explicitly asked}.$ After all, no one objects to "find all even perfect numbers" being a well-formed specific question (even though its answer involves Mersenne primes, which we don't completely know). Just to emphasize: I don't think we should go all the way back to Euclid, or to Pythagoreans, for that matter, because they already knew 6 and 28 were perfect numbers. | |
| Jun 4, 2010 at 22:17 | comment | added | Noah Snyder | So the reason I'm slightly uncomfortable going back all the way to Euclid is that "find all perfect numbers" is too imprecise a question to be called an open problem. Otherwise we could say "find all prime numbers" is a problem going back to whenever people started looking for them. "All perfect numbers are odd" or "there are infinitely many perfect numbers" are both well-formed specific problems. And certainly they were problems raised by Nicomachus as you can't (bogusly) answer a question without first having a question! | |
| Jun 4, 2010 at 21:12 | comment | added | Victor Protsak | Stretching it a bit further, we may ascribe the problem of finding perfect numbers to Euclid. He proved a formula for some of them, which is indirect evidence that he wanted to know them all. For the record, I disagree with this argument, but it's akin to crediting Nicomachus with the odd perfect number problem. | |
| Jun 4, 2010 at 21:04 | comment | added | Victor Protsak | Should we credit someone with posing X as an open problem if he made a claim implying that X is true? That's debatable. Dickson, History of theory of numbers, vol 1 says that Nicomachus classified $\textit{even}$ numbers into abundant, deficient and perfect and that he claimed that every perfect number is obtained by Euclid's rule. However, it's not clear that Nicomachus knew of existence of odd abundant numbers! So while we may speculate that Nicomachus has considered the question, as far as I can tell, there is no indication of it in his book. That makes the situation different from FLT. | |
| Jun 4, 2010 at 20:03 | comment | added | coudy | @Noah. Thanks for the link. It definitely answers my second question. | |
| Jun 4, 2010 at 20:03 | comment | added | Noah Snyder | Surely you would say that Fermat's Last Theorem (before it was solved) was an open problem dating back to Fermat! | |
| Jun 4, 2010 at 19:58 | comment | added | TonyK | Reading the link you provide about Nicomachus, it looks to me as if he stated that all perfect numbers are even. So he didn't regard it as an open problem. | |
| Jun 4, 2010 at 19:51 | history | edited | Noah Snyder | CC BY-SA 2.5 |
added 179 characters in body; added 61 characters in body
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| Jun 4, 2010 at 18:51 | history | answered | Noah Snyder | CC BY-SA 2.5 |