Question

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

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 mathamatics:

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

2) Gerneral 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 in a specilized field.

Bounty:

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

Related question: http://mathoverflow.net/questions/23982/where-have-you-used-computer-programming-in-your-career-as-an-applied-pure-math, http://mathoverflow.net/questions/68442/what-could-be-some-potentially-useful-mathematical-databases

Answer 1

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 $\sum_{p\leq x}\frac{1}{p}\sim \log\log x$.

Answer 2

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

Answer 3

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).