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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?

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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.

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    $\begingroup$ 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
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    $\begingroup$ @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
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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.

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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).

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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.

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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.

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  • $\begingroup$ I believe that holds some record for the shortest mathematical paper ever published? $\endgroup$ Jul 14, 2021 at 15:04
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    $\begingroup$ 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
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    $\begingroup$ 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? $\endgroup$
    – Voo
    Jul 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
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Four high school students here in the Netherlands constructed a $12\times12$ Franklin square with very extra symmetries. It wasn't completely Franklin though. The half horizontals didn't add up to half the whole value. I went on to develop a $16\times16$ completely symmetrical and magical $16\times16$ square. More magical (with the numbers $1-256$) it can't get. I made a $32\times32$ 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 $12\times12$ Franklin square (like the one the students made) doesn't exist. If perfect $12\times12$ Franklin squares existed was a riddle up till then.

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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.

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  • $\begingroup$ 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?" $\endgroup$
    – Mark
    Jul 19, 2021 at 3:35
  • $\begingroup$ But is it still a proof then? $\endgroup$ Jul 19, 2021 at 3:37
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    $\begingroup$ @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
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    $\begingroup$ @DescheleSchilder I do not dispute your view of computers. Be that as it may, until McCune used EQP to search for a proof of the Robbins conjecture, no human was able to prove it. But after studying the computer output, humans were then able to write down what everyone agrees is a correct proof of the Robbins conjecture. Dahn's paper is only 6 pages long and it is possible for a human to follow the reasoning one step at a time without any recourse to computers. But the motivations for the steps are not always clear. $\endgroup$ Jul 19, 2021 at 3:50
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If by "long-standing" we mean a decade or more, then this sort of thing is fairly common in combinatorics. Short, elementary arguments can be very difficult to find. Here are a few examples that I happen to know about; there are many more.

  1. The question of whether an aperiodic monotile or "einstein" exists has been implicitly around ever since Wang tiles were discovered in 1961. The first example was found in 2023, by Dave Smith, who was not even a professional mathematician. The proof that Smith's tile is an einstein is not that simple, but the bottleneck was finding a candidate tile in the first place, and the tile is very simple to describe (and was found by hand, not by a vast computer search).

  2. The union-closed sets conjecture, which was posed in 1979, is still open, but in 2022, Justin Gilmer made a huge breakthrough, proving a slightly weakened form of the conjecture using no concepts that were not around in 1979.

  3. Hao Huang's proof of the sensitivity conjecture was completely elementary but did not appear until 2019, whereas the conjecture was first posed in 1992.

  4. 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.

  5. The Stanley–Wilf Conjecture 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.

  6. Dvir proved the finite field Kakeya conjecture in 2008, about a decade after it was first posed, using a stunningly simple argument. No less a mathematician than Terry Tao had unsuccessfully thought about the conjecture. Dvir's use of the "polynomial method" paved the way for many other advances, such as the solution of the cap set problem, which was still quite elementary.

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The Gaussian correlation inequality was first conjectured in the 1950s, and not proved until 2014 by Thomas Royen. As Wikipedia puts it,

The proof did not gain attention when it was published in 2014, due to Royen's relative anonymity and the fact that the proof was published in a predatory journal. Another reason was a history of false proofs (by others) and many failed attempts to prove the conjecture, causing skepticism among mathematicians in the field. The conjecture, and its solution, came to public attention in 2017, when other mathematicians described Royen's proof in a mainstream publication and popular media reported on the story.

Quanta magazine describes the reaction of statistician Donald Richards as follows:

Upon seeing the proof, “I really kicked myself,” Richards said. Over the decades, he and other experts had been attacking the GCI with increasingly sophisticated mathematical methods, certain that bold new ideas in convex geometry, probability theory or analysis would be needed to prove it. Some mathematicians, after years of toiling in vain, had come to suspect the inequality was actually false. In the end, though, Royen’s proof was short and simple, filling just a few pages and using only classic techniques. Richards was shocked that he and everyone else had missed it.

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