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I would like to ask about examples where experimentation by computers have led to major mathematical advances.

A new look

Now as the question is five years old and there are certainly more examples of mathematical advances via computer experimentation of various kinds, I propose to consider contributing new answers to the question.


I am aware about a few such cases and I think it will be useful to gather such examples together. I am partially motivated by the recent polymath5 which, at this stage, have become an interesting experimental mathematics project. So I am especially interested in examples of successful "mathematical data mining"; and cases which are close in spirit to the experimental nature of polymath5. My experience is that it can be, at times, very difficult to draw useful insights from computer data.

Summary of answers according to categories

(Added Oct. 12, 2015)

To make the question a useful resource (and to allow additional answers), here is a quick summery of the answers according to categories. (Links are to the answers, and occasionally to an external link from the answer itself.)

1) Mathematical conjectures or large body of work arrived at by examining experimental data - Classic

The Prime Number Theorem; Birch and Swinnerton-Dyer conjectures; Shimura-Taniyama-Weil conjecture; Zagier's conjectures on polylogarithms; Mandelbrot set; Gosper Glider Gun (answer), Lorenz attractor; Chebychev's bias (asnwer) ; the Riemann hypothesis; the discovery of the Feigenbaum constant; (related) Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness conjecture (NEW); Solving numerically the so-called Fermi--Pasta--Ulam chain and then of its continuous limit, the Korteweg--de Vries equation

2) Mathematical conjectures or large body of work arrived at by examining experimental data - Current

"Maeda conjecture"; the work of Candès and Tao on compressed sensing; Certain Hankel determinants; Weari-Phelan structure; the connection of multiple zeta values to renormalized Feynman integrals; Thistlethwaite's discovery of links with trivial Jones polynomial; The Monstrous Moonshine; (NEW:) McKay's account on experimentation leading to mysterious "numerology" regarding the monster. (link to the answer); Haiman conjectures on the quotient ring by diagonal invariants

3) Computer-assisted proofs of mathematical theorems

Kepler's conjecture ; a new way to tile the plane with a pentagon: advances regarding bounded gaps between primes following Zhang's proof; Cartwright and Steger's work on fake projective planes; the Seifert-Weber dodecahedral space is not Haken; the four color theorem, the proof of the nonexistence of a projective plane of order 10; Knuth's work on a class of projective planes; The search for Mersenne primes; Rich Schwartz's work; The computations done by the 'Atlas of Lie groups' software of Adams, Vogan, du Cloux and many other; Cohn-Kumar proof for the densest lattice pacing in 24-dim; Kelvin's conjecture;

4) Computer programs that interactively or automatically lead to mathematical conjectures.


5) Various computer programs which allow proving automatically theorems or generating automatically proofs in a specialized field.

Wilf-Zeilberger formalism and software; FLAGTOOLS

6) Computer programs (both general purpose and special purpose) for verification of mathematical proofs.

The verification of a proof of Kepler's conjecture.

7) Large databases and other tools

Sloane's online encyclopedia for integers sequences; the inverse symbolic calculator.

8) Resources:

Journal of experimental mathematics; Herb Wilf's article: Mathematics, an experimental science in the Princeton Companion to Mathematics, genetic programming applications a fairly comprehensive website ; discovery and experimentation in number theory; Doron Zeilberger's classes called "experimental mathematics"; V.I. Arnol'd's two books on the subject of experimental mathematics in Russian, Experimental mathematics, Fazis, Moscow, 2005, and Experimental observation of mathematical facts, MCCME, Moscow, 2006

Answers with general look on experimental mathematics:

Computer experiments allow new avenues for productive strengthening of a problem (A category of experimental mathematics).


There were many excellent answers so let's give the bounty to Gauss...

Related question: Where have you used computer programming in your career as an (applied/pure) mathematician?, What could be some potentially useful mathematical databases?; Results that are easy to prove with a computer, but hard to prove by hand ; What advantage humans have over computers in mathematics?

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8… – Steve Huntsman Jan 17 '10 at 8:35
You might want to email Doron Zeilberger at Rutgers. – Charles Siegel Jan 17 '10 at 15:39
Indeed, Doron Zeilberger has even taught classes called "experimental mathematics": . – Qiaochu Yuan Jan 17 '10 at 15:50
@Mark: URLs are automatically turned into links in comments; no need for html markup. Corrected version of the link: – Anton Geraschenko Jan 20 '10 at 18:53
Doron has recently posted a "videoed lecture" about experimental mathematics: – Douglas S. Stones Apr 3 '10 at 0:48

42 Answers 42

The solution of the Kepler's conjecture by Hales deserves a special answer since it gives an important example both for a proof to a mathematical theorem massively using computers, and 15 years later by an impressive project for demonstrating a formal proof for the conjecture that could be automatically verified.

The problem and the first project which in part was carried out by Hales with Ferguson is described in the Notices AMS paper "Cannonballs and Honeycombs" Another reference is a 2002 ICM survey A Computer Verification of the Kepler Conjecture.

The published account of the second project - The Flyspeck project can be found in the paper A formal proof of the Kepler conjecture.

For more information and links, see Hales homepage. enter image description here

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Thurston's geometrization program began with Bob Riley's computer experiments - Thurston's intuition was actually initially the opposite (he did not believe that the figure eight knot could be hyperbolic until he saw the experimental result).

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wow! I did not know that... – Gil Kalai Sep 3 '15 at 12:49

In a remarkable article from 1989, "Self-similarity and hairiness in the Mandelbrot set", Milnor formulates a number of conjectures (related to Feigenbaum universality). Citing Lisa Goldberg and Tony Phillips in the preface to Topological Methods in Modern Mathematics (Milnor's 60th birthday Festschrift, 1993),

OF special note is an unconventional article entitled Self-similarity and hairiness in the Mandelbrot set, where he presents some of his numerical experiments in holomorphic dynamics and uses them as evidence for a set of conjectures. ... The resulting data are so compelling as to suggest not only what the conjectures should say, but how they can be proved.

Lyubich later proved some of these conjectures in a groundbreaking article (Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness conjecture, Annals of Mathematics, 1999), which also, for the first time, gave a conceptual proof of the Feigenbaum scaling law.

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(Answer by John Mckay:) That the product of two conjugate (type 2A) involutions of the monster, M lies in 9 conjugacy classes with order given by the coefficients of the highest root in affine E8, namely {1,2,3,4,5,6,4,2,3} -- and similarly for E7 and E6 with 2.Baby and 3.F24' respectively -- was experimental with very little evidence. See my note in Nature 305, 672, 20th Oct 1983 on numerology.

(GK: Indeed this is a very important example in the history of mathematics. Here is the Nature's paper)
enter image description here

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A fake projective plane is a smooth compact complex surface with the same Betti numbers as $\mathbb{P}^2(\mathbb{C})$. Mumford (MR0527834) built the first one in the late 70s using $p$-adic uniformization. Klingler (MR1990668) and, independently, Yeung (MR2128300 with erratum MR2559112), showed that they necessarily corresponded to arithmetic lattices in $\mathrm{PU}(2,1)$. (That they live in $\mathrm{PU}(2,1)$ follows from Yau's Theorem MR0451180; they do the arithmeticity.) Prasad and Yeung (MR2289867) then gave the list of arithmetic constructions which could possible create such a beast.

Using a computer experiment, Cartwright and Steger gave presentations for all the fundamental groups, giving a complete classification. They have recent announcement in Comptes Rendus #348, 11-13. Their presentations also allow one to show that the congruence subgroup property doesn't hold for certain commensurability classes of lattices for which it was previously unknown. In fact, these lattices are a few of the `simplest' examples for which the congruence subgroup problem remained mysterious. (Serre conjectures in his paper on $\mathrm{SL}_2$, MR0272790, that it fails for all $\mathbb{R}$-rank one groups.)

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I think there may be an additional category, though relatively little-known outside my own field (design automation): genetic programming applications, or more generally the application of metaheuristics. Spector et al's "Genetic Programming for Finite Algebras" (2008) is a good example of the methodology, which seems to me to be much less about the automation of proof as such, and more about enhancing mathematicians' capacity for well-defined combinatorial searches.

That said, you may want to include it, and the growing number of other GP papers in this domain, in your survey categories under (4) or (5).

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In some simple cases it is even possible to both guess some results and prove them too experimentally. I have done this with some Hankel determinants. Suppose you want to compute the Hankel determinant $\det \left( {a_{i + j} } \right)_{i,j = 0}^{n - 1} $ of a sequence $(a_n )$ . If no Hankel determinant vanishes then you can compute polynomials $p_n (x)$ which are orthogonal with respect to the linear functional $F$ on the polynomials defined by $F(x^n ) = a_n .$ By Favard's theorem there exist numbers $s_n ,t_n $ such that $p_{n + 2} (x) = (x - s_{n + 1} )p_{n + 1} (x) - t_n p_n (x).$ In some cases it is easy to guess a formula for these numbers after computing some of them. Thus also the Hankel determinant can be computed. In order to show that this guess gives the correct answer we can do the following: Define $a(n,j) = a(n - 1,j - 1) + s_j a(n - 1,j) + t_j a(n - 1,j + 1)$ with $a(0,j) = \delta _{j,0} .$ By the underlying theory it suffices to show that $a(n,0) = a_n .$ In order to do this we again compute $a(n,j)$ for small values and try to guess a closed formula for them. In many cases we succeed. Then it suffices to verify that $a(n,j) = a(n - 1,j - 1) + s_j a(n - 1,j) + t_j a(n - 1,j + 1)$ holds for the conjectured formula.

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Huh, wasn't there already a known recurrence for computing successive Hankel determinants? See for instance page 8 of – J. M. Sep 3 '10 at 3:28
Thank you for the interesting link about recurrences for Hankel determinants. But my remark has been about experimentally finding and proving explicit formulas for some classes of Hankel determinants as in – Johann Cigler Sep 3 '10 at 7:40

The Shimura-Taniyama-Weil conjecture (now a theorem) and the Birch and Swinnerton-Dyer conjectures were based on examples.

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I think that Zagier's conjectures on polylogarithms were based on considerable amount of numerical evidence (this has led to impressive work by Beilinson-Deligne, Goncharov...).

Serre's conjecture on modularity of mod $p$ $2$-dimensional Galois representations was also made precise thanks to simultaneous theoretical advances and numerical computations (this has paved the way to the proof of Fermat's last theorem...).

To come back to the prime number theorem, I think that Euler already "proved" it long before Gauss conjectured it by differentiating $\sum_{p\leq x}\frac{1}{p}\sim \log\log x$.

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Hida and Maeda's article "Non-abelian base change for totally real fields" in the pacific journal of mathematics, volume 181(3) in 1997 explains how the experimental data lead to the "Maeda conjecture" (which you'll find mentioned in a few questions on MO).

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Graffiti is a computer program which makes conjectures in various subfields of mathematics, particularly graph theory. (And also in chemistry.) It was written by written by Siemion Fajtlowicz. Here is a 1989 article about it.

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First, a brief story, then the punchline.

Elsewhere I mention a series of 0-1 matrices used for a combinatorial matrix result. This came about because of an aspiring undergraduate (Roger House) who was using a computer to search for 0-1 matrices of certain orders and determinant values.

He showed me a printout where he listed the first representative of each class (major key order, minor key absolute determinant value). He chose a lexicographical ordering of class members to pick this representative: this choice revealed a set of matrices with few differing entries, and those in just the first two or three columns. I would not blame someone else for saying it was a pattern screaming to be found.

Many a time since then, I think about how to organize computer-generated data (specifically, pick an explanatory set of representatives) that reveal patterns and structure. I think that experimental mathematics should not just include results, or even data analysis techniques, but heuristics on "how to make choices" so that patterns of note jump to the human eye. In the case above, lexicographic order was a choice made by one person that revealed a pattern of similarity to another person.

Gerhard "At Least, Raise Their Voices" Paseman, 2015.10.12

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