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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.

Motivation

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.

Meta Question: classification of experimental mathematics

I asked: Is there a useful way to classify the various types of experimental mathematics?

Let me try to propose, based on the answers so far, a tentative answer describing five categories of experimental mathematics:

1) Mathematical conjectures that were arrived at by examining experimental data

2) General purpose programs which interactively or automatically lead to posing mathematical conjectures.

3) Large mathematical databases

4) Computer-assisted proofs of mathematical theorems

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

(Two additional categories added, Aug 2015)

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

7) (Beyond the scope of this question) Traditional applications of computers in various areas of applied mathematics - numerics, modeling, simulations, etc.

Bounty:

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?

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

36 Answers 36

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

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Some of the advances regarding bounded gaps between primes following Zhang's proof were using computers and computations quite substantially. This applies to progress in polymath8, to Meynard's paper, and to progress in polymath8b. See this paper for a general perspective of these advances.

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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 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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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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