# Examples of simultaneous independent breakthroughs

I'm looking for examples where, after a long time with little progress, a simultaneous mathematical discovery, solution, or breakthrough was made independently by at least two different people/groups. Two examples come to mind:

1. Prime Number Theorem. This was proved by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896.
2. Sum of three fourth powers equals a fourth power. In 1986, Noam Elkies proved that there are infinitely many integer solutions to $$a^4 + b^4 + c^4 = d^4$$. His smallest example was $$2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$$. Don Zagier reported that he found a solution independently just weeks later.

Can you give other instances?

• This happens frequently. Search for the word "independently" in abstracts or in the conclusion (or introduction) of papers. My name appears in Algebra Universalis because Libor Polak acknowledged my near simultaneous solution to a problem in logic he solved. (In this case though, the problem had not been around long.) Gerhard "Or Look Up 'On Hyperassociativity'" Paseman, 2019.07.26. – Gerhard Paseman Jul 26 '19 at 19:31
• Having studied the case $a^4+b^4+c^4 = 1$ and the case $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$, the Zagier anecdote was interesting to know. – Tito Piezas III Jul 28 '19 at 2:54
• There was some discussion of why there are simultaneous independent discoveries at math.stackexchange.com/questions/709969/… – Gerry Myerson Jul 30 '19 at 4:21

One of the most entertaining seminar talks I ever attended was by Michael Atiyah on the moment map on a Lie group. As the talk progressed, the people in the front row became more and more agitated, until Raoul Bott finally spoke up, said something like, "Michael, these guys just proved the same theorem last week!", and pointed to Victor Guillemin and Shlomo Sternberg.

The Gelfond-Schneider Theorem, if $$a$$ and $$b$$ are algebraic numbers with $$a\ne0,1$$, and $$b$$ irrational, then any value of $$a^b$$ is a transcendental number, was proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.

One of the most famous ones seems to be Newton vs. Leibniz, see e.g. https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy https://www.springer.com/de/book/9783319725611

• Neither was solving a problem, however. Arguably what both were doing was to turn various ideas that had been around for a while into a “method”. A breakthrough nonetheless, I suppose. – Fernando Jul 30 '19 at 22:40

The solution of Hilbert's nineteenth problem, in 1957, by Ennio De Giorgi and John Nash, few months later.

The near simultaneous solution to Serre's conjecture on projective modules over polynomial rings by Quillen and Suslin independently comes to mind.

Recent example in analytic number theory:

http://annals.math.princeton.edu/2016/183-3/p03

and

http://annals.math.princeton.edu/2016/183-3/p04

• Please give more than just two links. – David Richerby Jul 29 '19 at 15:16
• @R.vanDobbendeBruyn the specific problem they solved have remain virtually untouched since the 1960s. What Erdos proposed was very modest: can one prove that there exist infinitely may primes $p$ such that the gap between $p$ and the next prime $p^\prime$ is at least $c \cdot g(p)$, where $g$ is a very specific function and $c$ an arbitrarily large positive number. Both sets of authors accomplished this and released their results within a week of each other. – Stanley Yao Xiao Jul 29 '19 at 19:12
• It's true that these two papers were independent of each other, but they both relied heavily on Yitang Zhang's breakthrough, which had happened not long before. Somehow this makes the papers seem less independent than some of the other examples in this list. – Timothy Chow Aug 21 '19 at 20:22
• @TimothyChow the story is a bit convoluted. When Maynard announced his proof of bounded gaps between primes, there was much speculation as to whether or not he already had the key ideas before Zhang surprisingly announced his proof six months prior. I talked to James directly and he confirmed to me that he did not, as of May 2013, believed he had a proof of the result, but was inspired by Zhang's work to refine an earlier method of his which ultimately turned into a proof of bounded gaps. Tao gave a similar idea on his blog around the same time as Maynard's paper appeared on arxiv. – Stanley Yao Xiao Aug 21 '19 at 23:18
• One can look at Maynard's earlier work, which is now published as sciencedirect.com/science/article/pii/S0022314X15000657, and confirm that he has essentially developed his sieve independently of Zhang's work. This is also clear if one is familiar with how Zhang and Maynard's proofs work... they are quite different. Zhang's key idea was that a restricted improvement to the Bombieri-Vinogradov theorem, which gives a slightly larger level of distribution, could be proved and combined with the earlier work of Goldston-Pintz-Yildirim, would give bounded gaps. – Stanley Yao Xiao Aug 21 '19 at 23:23

Abel and Galois were working independently and did not know each other. Their approaches were different but they solved (among other things) an old and famous problem.

There are many other examples: Abel and Jacobi (elliptic functions), Bolyai, Lobachevski and Gauss independently developed non-Euclidean geometry.

• Will it be correct to say that Abel and Galois solved the same problem? – Sergei Akbarov Jul 27 '19 at 13:46
• @Sergei Akbarov: Yes, besides other things they proved unsolvability of a general equation of degree >4 in radicals. – Alexandre Eremenko Jul 27 '19 at 15:27
• I thought it was Abel who proved this... Didn't Galois know this? – Sergei Akbarov Jul 27 '19 at 15:40
• Galois was aware of Abel's work: doi.org/10.1007%2FBF00357046 – Robert Furber Jul 27 '19 at 18:20
• Galois knew Abel’s work and solved a different problem as well. Plus, there is a distance of about 10 years. – Fernando Jul 30 '19 at 22:32

A very recent example appears to be the simultaneous discovery of functorial (and "easy") resolution of singularities in characteristic 0 by McQuillan-Marzo (arXiv:1906.06745) respectively Abramovich-Temkin-Włodarczyk (arXiv:1906.07106).

I'm not sure if it satisfies the criteria of "a long time with little progress", but one example of a simultaneous mathematical discovery is the discovery of the hook length formula. Here is how Bruce Sagan describes it in his book The Symmetric Group:

One Thursday in May of 1953, Robinson was visiting Frame at Michigan State University. Discussing the work of Staal (a student of Robinson), Frame was led to conjecture the hook formula. At first Robinson could not believe that such a simple formula existed, but after trying some examples he became convinced, and together they proved the identity. On Saturday they went to the University of Michigan, where Frame presented their new result after a lecture by Robinson. This surprised Thrall, who was in the audience, because he had just proved the same result on the same day!

Richard Friedberg and Albert Muchnik independently and almost simultaneously solved Post's problem.

Ramanujan graphs were independently constructed by Margulis and by Lubotzky–Phillips–Sarnak.

A conjecture from the 1950s (arising from a result of Kaplansky) was that for any compact Hausdorff space $$X$$, every algebra homomorphism from $$C(X)$$ into another Banach algebra is automatically continuous. This conjecture was refuted in the late 1970s, independently, by H. Garth Dales and Jean Esterle: a very short announcement which outlines some of the ideas can be found in

H. G. Dales and J. Esterle, Discontinuous homomorphisms from C(X). Bull. Amer. Math. Soc. 83 (1977), 257–259

Both approaches assumed the continuum hypothesis; later work of Solovay exhibited models of ZFC in which the original conjecture has a positive answer.

The theoretical basis of Linear Optimization has been independently developed in the time of the second world war by L.V. Kantorovich and by G.B. Dantzig.

Van der Waerden's conjecture on the permanent of a doubly stochastic matrix had been open for 50 years when it was proved independently by Falikman and Egorychev.

The independence of the Parallel Postulate had been open for about 2000 years when its independence had been independently by Bolyai, Lobachevski and Gauss around 1830; Gauss never published his proof, but only claimed to have solved the problem when he was informed about Bolyai's proof.

• That said, there is no doubt that Gauss was very much occupied with this question since the time he was a young teenager, and he spoke in some detail about his ideas regarding this question in letters to others. So I think it's a case of more than "only claiming". – Todd Trimble Jul 27 '19 at 13:53
• @ToddTrimble I was only being careful with my formulation, taking the point of view of Janos Bolyai at that time; Janos Bolyai had suspected that Gauss had 'stolen' his ideas which he assumed his father Farkas had mentioned in letters to Gauss . The English wikipedia article doesn't mention that, whereas the German wikipedia article mentions that Janos developed a serious illness after Gauss said to already have made the discovery. I also think that Gauss was everything else but dishonest. – Manfred Weis Jul 27 '19 at 15:15
• Gauss, true to form, did not behave all that graciously with regard to Bolyai, saying in a letter to Farkas Bolyai (formerly a schoolmate of Gauss), in effect, "I cannot bring myself to praise this accomplishment of your son Janos, because to do so would be praising myself". He ends that letter on a slightly better note, but there is no doubt that the letter was received as a wet blanket on the young man's enthusiasm. But "stealing" is wrong: Gauss had already worked out detailed calculations on hyperbolic geometry and had satisfied himself that it was wholly consistent. – Todd Trimble Jul 27 '19 at 18:11
• I cannot make out what you are alleging in your final sentence, "I also think that Gauss was everything else but dishonest." – Todd Trimble Jul 27 '19 at 18:12
• I thought the independence of the Parallel Postulate (i.e. the consistency of hyperbolic geometry) was proved by Beltrami in 1868? – bof Jul 28 '19 at 12:51

Recently I reviewed

Shmerkin, Pablo On Furstenberg's intersection conjecture, self-similar measures, and the $$L^q$$ norms of convolutions. Ann. of Math. (2) 189 (2019), 2, 319--391

and obverved that another proof of Furstenberg's intersection conjecture is given in

Wu, Meng, A proof of Furstenberg’s conjecture on the intersections of ×p and ×q-invariant sets, Ann. of Math. (2) 189, 3, 707-751, (2019)

The methods of the proofs seems to be quite different.

The theory of NP-completeness was simultaneously and independently developed in North America and the U.S.S.R.

• Was Cook still in the U.S. when he proved NP-completeness of SAT? – Andreas Blass Jul 27 '19 at 14:55
• I quote the following from an article by Richard Karp: "Steve Cook was primarily in math but also in the new CS Department. It is to our everlasting shame that we were unable to persuade the math department to give him tenure. Perhaps they would have done so if he had published his proof of the np-completeness of satisfiability a little earlier." (from www2.eecs.berkeley.edu/bears/CS_Anniversary/karp-talk.html ) This would seem to indicate that Cook didn't publish the NP-completeness of SAT until he was in Toronto. – Robert Furber Jul 28 '19 at 1:55
• The paper itself also associates Cook with the University of Toronto. (cs.toronto.edu/~sacook/homepage/1971.pdf) – nwn Jul 28 '19 at 13:10
• OK but the answer claims he developed it in the US and makes no claims about where he was by the time it was published. – David Richerby Jul 29 '19 at 15:39
• @DavidRicherby That's why I asked about "when he proved" rather than "when he published". – Andreas Blass Jul 29 '19 at 18:51

The "class number one" problem for imaginary quadratic fields was posed by Gauss in 1798, settled independently by Baker and Stark in 1966/7. Gauss had found nine imaginary quadratic fields with class number one (which, in this context, is equivalent to unique factorization into primes) and conjectured there were no others; Baker and Stark proved this.

(There was also a proof by Heegner some years earlier than the work of Baker and Stark, but this proof had some minor gaps and was not generally accepted until Stark showed how to fill the gaps in 1969.)

The equivalence of nondeterministic logarithmic complexity class (NL) to the class whose complements are in NL (co-NL) was proven independently by Neil Immerman (University of Massachusetts, Amherst, MA, USA) and Róbert Szelepcsényi (Comenius University, Bratislava, Slovakia). This result is known as the Immerman–Szelepcsényi theorem and they shared the 1995 Gödel Prize for it.

During the summer of 2001, Crew's conjecture in the theory of $$p$$-adic differential equations was simultaneously proved in three very different ways by André, Kedlaya, and Mebkhout.

The word problem for groups was shown to be undecidable by Pyortr Novikov in 1955 and William Boone in 1958. Since there was practically no academic communication between their home countries (USSR and USA, respectively) at the time, the consensus is that the discoveries were independent.

• Independent, yes, but does it qualify as "simultaneous", as requested by OP? – Gerry Myerson Jul 29 '19 at 23:16

In 1992, Feder and Vardi conjectured that for every constraint language $$\Gamma$$, the constraint satisfaction problem with constraints from $$\Gamma$$ is either solvable in polynomial time or is $$\mathrm{NP}$$-complete. Progress was made on special cases and related problems but not a whole lot happened until Bulatov and Zhuk independently announced proofs of the conjecture in March and April 2017.

As I recall, a third proof was announced at around the same time, but was subsequently found to be erroneous and withdrawn.

Obviously, the most famous example would be Newton-Leibniz development of Calculus, but there is also the case of Lagrange and Euler when developing the calculus of variations. A list can be found here.

• I think Lagrange knew Euler’s work. – Fernando Jul 30 '19 at 22:35
• Newton-Leibniz was mentioned by another user a couple of days earlier than this post. – Gerry Myerson Oct 31 '20 at 23:55

The HOMFLY or HOMFLYPT polynomial is a clear example. The polynomial was introduced in "A new polynomial invariant of knots and links", and on the first page there is a footnote:

Editor's Note. The editors received, virtually within a period of a few days in late September and early October 1984, four research announcements, each describing the same result—the existence and properties of a new polynomial invariant for knots and links. There was variation in the approaches taken by the four groups and variation in corollaries and elaboration. These were: A new invariant for knots and links by Peter Freyd and David Yetter; A polynomial invariant of knots and links by Jim Hoste; Topological invariants of knots and links, by W. B. R. Lickorish and Kenneth C. Millett, and A polynomial invariant for knots: A combinatorial and an algebraic approach, by A. Ocneanu.

It was evident from the circumstances that the four groups arrived at their results completely independently of each other, although all were inspired by the work of Jones (cf. [10], and also [8, 9]). The degree of simultaneity was such that, by common consent, it was unproductive to try to assess priority. Indeed it would seem that there is enough credit for all to share in.

Each of these papers was refereed, and we would have happily published any one of them, had it been the only one under consideration. Because the alternatives of publication of all four or of none were both unsatisfying, all have agreed to the compromise embodied here of a paper carrying all six names as coauthors, consisting of an introductory section describing the basics written by a disinterested party, and followed by four sections, one written by each of the four groups, briefly describing the highlights of their own approach and elaboration.

The result also appears in work by Józef H. Przytycki and Paweł Traczyk, hence the PT. The reason for simultaneous discovery is quite clear: the HOMFLYPT polynomial is a relatively straightforward generalization of the (at the time) recently introduced Jones polynomial.

The angel problem was posed in 1982, and little progress was made until it was solved independently and almost simultaneously in 2006 by four different people. To borrow from the Wikipedia article, the question is to determine the winner of a certain game:

The game is played by two players called the angel and the devil. It is played on an infinite chessboard. The angel has a power $$k \in \mathbb{N}$$ specified before the game starts. On each turn, the angel flies to a different square whose distance from its current square is at most $$k$$ in the infinity norm. The devil, on its turn, may permanently block any single square not containing the angel. The angel may fly over blocked squares, but cannot land on them. The devil wins if the angel is unable to move. The angel wins by surviving indefinitely.

The problem was first published in the book Winning Ways for Your Mathematical Plays by Berlekamp, Conway, and Guy and therefore became fairly well known among people interested in recreational mathematics. However, arguably the reason for the sudden appearance of four independent solutions in 2006 was the publication of Peter Winkler's book Mathematical Puzzles: A Connoisseur's Collection, which listed the angel problem as an unsolved problem, and presumably sparked the interest of a lot of people who had either not heard the problem before, or at least had not given it much thought.

The Entscheidungsproblem was a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms.

The problem was solved independently by Alonzo Church and A. M. Turing in 1936. Church's solution was based on lambda calculus, whereas Turing's approach was based on hypothetical computational devices now known as Turing Machines.

On a related note, a definition of computation was also provided independently by Emil L. Post in 1936, and is equivalent to the the other two approaches.

Hillel Furstenberg and Robert Zimmer simultaneously proved a powerful structure theorem for measure preserving dynamical systems: exposition at Tao’s blog. Furstenberg’s article also contains a proof of a conjecture of László Lovász, on the density of a set of integers, no two of which differ by a perfect square. This was proved simultaneously and independently with a different method, by András Sárközy.

Ray and Singer conjectured that their analytic torsion should carry the same information as Reidemeister torsion. This was then proved independently by Cheeger and Müller, see wikipedia.

But maybe this does not count because the phase of inactivity was not that long ...

• I think the setting for Cheeger and Muller's proof is slightly different. Cheeger's proof made some hypothesis which is not in original Ray-Singer paper. But this is based on my memory; I have not checked the paper for a long time. – Bombyx mori Jul 27 '19 at 10:13

A completely new algorithm for calculating $$\pi$$ was independently discovered by Salamin and Brent in 1976. This iterative algorithm, which doubles the number of correct digits with each iteration, is based on work of Gauss (arithmetic-geometric means) that had been neglected for 170 years.

• I quite disagree with the "neglected for 170 years"characterization; people have used AGM to compute elliptic integrals and functions way before the Brent-Salamin proposal (e.g. King (1924), Bartky (1938), Salzer (1962), Bulirsch (1965)). Relatedly, the Legendre relation on which Brent-Salamin is built on was presented in the 1800s, around the same time Gauss started investigating the AGM. – J. M. isn't a mathematician Jul 28 '19 at 14:18
• This answer would better suit mathoverflow.net/questions/176425/… – YCor Jul 28 '19 at 21:19

Last month (September 2020) Chodosh and Li uploaded a preprint showing that a closed aspherical manifold of dimension $$4$$ or $$5$$ does not admit a Riemannian metric with positive scalar curvature.

On exactly the same day Gromov uploaded a preprint which used similar ideas to generalize the result of Chodosh and Li to higher dimension (again, as usual, the fact that they were both submitted on the same day was almost certainly coordinated between the authors and not a coincidence).