Famous conjectures whose solutions took decades or centuries were usually resolved with the help of sophisticated theories and techniques unknown at the time the conjecture was first claimed. Is there any example where mathematicians didn't see a solution that after the fact was recognized as simple, basic or not very elaborated but people failed to see for a long time?
Is there any example of a long-standing mathematical conjecture whose resolution did not require advanced knowledge?
Sylvester's line problem (1893) was to prove that there exists no finite configuration of points in real projective plane such that every line through two points actually contains at least 3 points, unless all points are on the same line. Sylvester himself found such a configuration in the complex projective plane.
This problem was quite famous, and was considered very difficult: people compared its difficulty with the 4-color problem.
It was solved in 1944 by Grunwald, and a simple solution was obtained by Kelly in 1948. This solution occupies less than 1 page, and can be understood by any high school student.
Edit. Another somewhat similar example is "Gehring's problem on linked curves" described in my answer to: this question.
Second edit. The so called BMV conjecture was proposed in 1975, a lot of sophisticated mathematics was used in attempts to prove it. It was proved by H. Stahl in 2011, and then I simplified his proof to 6 pages using nothing that was not well-known in 1900.
5$\begingroup$ This may be a stupid question, but you wrote "Sylvester's line problem" and then "Silvester himself". Did you mean the same mathematician both times? Which is the correct spelling? $\endgroup$ Jul 12, 2021 at 18:24
$\begingroup$ I think this is discussed in more detail in en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem $\endgroup$ Jul 12, 2021 at 20:29
2$\begingroup$ @Ross Presser: Yes, I meant John Joseph Sylvester. I corrected the misprint, thanks. But Grunwald is the same person as Gallai:-) $\endgroup$ Jul 13, 2021 at 14:35
Hanna Neumann conjecture was open for 40+ years and was solved in 2011 by relatively elementary methods independently and almost simultaneously by Friedman and Mineyev. The existing proofs actually proved the strengthened Hanna Neumann conjecture formulated in 1990 by Walter Neumann. All tools used in Mineyev's proof certainly existed in 1990 and much earlier.
Just this year the 80 years old Kaplansky unit conjecture was disproved by Giles Gardam, also by quite elementary methods.
Tiling of the plane by pentagons.
After the problem was discussed in Martin Gardner's "Mathematical Games" column in Scientific American magazine of July 1975, Marjorie Rice (an amateur mathematician) discovered four new examples.
Euler's conjecture stated in 1769 https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture was resolved as disproven in 1966 using one basic line
$$27^5 + 84^5 +110^5 + 133^5= 144^5$$
https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf There may be more similar cases from that time-period of counter-examples that are trivial to check, but hard to find.
$\begingroup$ I believe that holds some record for the shortest mathematical paper ever published? $\endgroup$ Jul 14, 2021 at 15:04
4$\begingroup$ Related MO question on how they might have found this: mathoverflow.net/questions/325192/…. There's a second paper that spells out how it was found. $\endgroup$ Jul 14, 2021 at 16:43
2$\begingroup$ Given that finding this counter sample required the use of a mainframe (something which wasn't available in Euler's time), does this qualify though? $\endgroup$– VooJul 14, 2021 at 20:09
$\begingroup$ @Voo It depends on how you view it, checking it didn't require a mainframe - but finding it would have been tedious without such help. $\endgroup$ Jul 15, 2021 at 7:25
Perhaps the poster child for this kind of thing is Apéry's theorem that ζ(3) is irrational, as described for example in Alf van der Poorten's paper, A proof that Euler missed (Mathematical Intelligencer 1 (1979), 195–203). Apéry's proof was highly ingenious and original, and not easy to grasp even after the fact, but it was short and used only techniques that Euler would have had no difficulty understanding (hence the title of Van der Poorten's paper).
Four high school students here in the Netherlands constructed a 12x12 Franklin square with very extra symmetries. It wasn'f completely Franklin though. The half horizontals didn't add up to half the whole value. I went on to develop a 16x16 completely symmetrical and magical 16x16 square. More magical (with the numbers 1-256) it can't get. I made a 32x32 one too. With paper, pencil, and a calculator.
After they were on the news, a considerable wind was blowing in the magic square country. It was proven by direct calculation that a 12x12 Franklin square (like the one the students made) doesn't exist. If perfect 12x12 Franklin squares existed was a riddle up till then.
The Robbins conjecture was posed in the 1930s but not proved until 1996. The proof was achieved by William McCune with the assistance of the automated theorem prover EQP. Unlike some other computer-assisted proofs, this proof was digestible by a human being, especially after simplifications found by Bernd Dahn.
Dahn's version of the proof is short and completely elementary, but still rather mysterious. It shows that a certain identity is a formal consequence of the axioms, but via a sequence of manipulations that is difficult for a human to find, not so much because the proof is infeasibly long or laborious, but because there does not seem to be a clear conceptual idea to guide one's thinking from beginning to end. The proof is simple enough that it could have been discovered without computers, but at the same time, it is not too surprising that it was first discovered with the aid of a computer.
$\begingroup$ If there is no guiding concept then how can it be followed? $\endgroup$ Jul 18, 2021 at 21:12
$\begingroup$ @DescheleSchilder, most proofs have an internal logic to them: it's obvious that each step gets you closer to the desired target. This one, presumably, has one or more steps where the reader will go "why on Earth would you apply that theorem to this situation?" $\endgroup$– MarkJul 19, 2021 at 3:35
$\begingroup$ But is it still a proof then? $\endgroup$ Jul 19, 2021 at 3:37
1$\begingroup$ @DescheleSchilder One can verify that each step follows logically from what has come before, but the motivation for taking the step is not always so clear. Many proofs that are discovered by computers have this flavor. $\endgroup$ Jul 19, 2021 at 3:38
$\begingroup$ Thats because a computer has no motivations. A computer doesnt truly discover like it cant truly play check. It just tries zillions of approaches and if one fits it fits, like a move in chess. $\endgroup$ Jul 19, 2021 at 3:43
Maybe 15 years does not deserve to be called "long-standing," but the Dinitz conjecture was posed around 1979 and not proved until 1994, by Fred Galvin, by a completely elementary argument. There had been some serious attempts to prove the Dinitz conjecture, e.g., by Jeannette Jansen, so it wasn't as though nobody knew about the conjecture.
Another example is the Stanley–Wilf Conjecture, which also lasted about 15 years before being proved by Marcus and Tardos, by an argument that could in principle have been given as soon as the conjecture had been posed.
By the way, if you ever meet someone who thinks that combinatorics is trivial, you can try challenging them to prove these results (of course, without cheating and looking up the answer). Neither the theorem nor the proof requires a lot of machinery, but very few people are able to find the proof on their own.