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

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

#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.

##Summary of answers according to categories (Added Oct. 12, 2015)

###Bounty:

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.

Summary of answers according to categories

(Added Oct. 12, 2015)

Bounty:

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; Chebyshev's bias (answer) ; the Riemann hypothesis; the discovery of the Feigenbaum constant; (related) Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness conjecture; Solving numerically the so-called Fermi--Pasta--Ulam chain and then of its continuous limit, the Korteweg--de Vries equation

"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; 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

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 experimentalmath.info ; discovery and experimentation in number theory; Doron Zeilberger's classes called "experimental mathematics":math.rutgers.edu/~zeilberg/teaching.html; 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

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; Chebyshev's bias (answer) ; the Riemann hypothesis; the discovery of the Feigenbaum constant; (related) Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness conjecture; Solving numerically the so-called Fermi--Pasta--Ulam chain and then of its continuous limit, the Korteweg--de Vries equation

"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; 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

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 experimentalmath.info ; discovery and experimentation in number theory; Doron Zeilberger's classes called "experimental mathematics":math.rutgers.edu/~zeilberg/teaching.html; 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