Editing Experimental mathematics (section)

==Applications and examples==
Applications and examples of experimental mathematics include:

*Searching for a counterexample to a conjecture
**Roger Frye used experimental mathematics techniques to find the smallest counterexample to [[Euler's sum of powers conjecture]].
**The [[ZetaGrid]] project was set up to search for a counterexample to the [[Riemann hypothesis]].
**Tomás Oliveira e Silva<ref>{{cite web|first=Tomás|last=Silva|website=Institute of Electronics and Informatics Engineering of Aveiro|title=Computational verification of the 3x+1 conjecture|url=http://sweet.ua.pt/tos/3x+1.html|archive-url=https://web.archive.org/web/20130318045112/http://www.ieeta.pt/~tos/3x+1.html|archive-date=18 March 2013|date=28 December 2015|url-status=live}}</ref> searched for a counterexample to the [[Collatz conjecture]].
*Finding new examples of numbers or objects with particular properties
**The [[Great Internet Mersenne Prime Search]] is searching for new [[Mersenne prime]]s.
**The Great Periodic Path Hunt is searching for new periodic paths.
**[[distributed.net]]'s OGR project searched for optimal [[Golomb ruler]]s.
**The [[PrimeGrid]] project is searching for the smallest [[Riesel number|Riesel]] and [[Sierpiński number|Sierpiński]] numbers.
*Finding serendipitous numerical patterns
**[[Edward Lorenz]] found the [[Lorenz attractor]], an early example of a chaotic [[dynamical system]], by investigating anomalous behaviours in a numerical weather model.
**The [[Ulam spiral]] was discovered by accident.
**The pattern in the [[Ulam number]]s was discovered by accident.
**[[Mitchell Feigenbaum]]'s discovery of the [[Feigenbaum constant]] was based initially on numerical observations, followed by a rigorous proof.
*Use of computer programs to check a large but finite number of cases to complete a [[computer-assisted proof|computer-assisted]] [[proof by exhaustion]]
**[[Thomas Callister Hales|Thomas Hales]]'s proof of the [[Kepler conjecture]].
**Various proofs of the [[four colour theorem]].
**[[Clement Lam]]'s proof of the non-existence of a [[projective plane|finite projective plane]] of order 10.<ref>{{cite journal |author=Clement W. H. Lam |title=The Search for a Finite Projective Plane of Order 10 |journal=[[American Mathematical Monthly]] |volume=98 |issue=4 |year=1991 |pages=305–318 |url=http://www.cecm.sfu.ca/organics/papers/lam/ |doi=10.2307/2323798|jstor=2323798 |url-access=subscription }}</ref>
**Gary McGuire proved a minimum uniquely solvable [[Sudoku]] requires 17 clues.<ref>{{cite news|last1=arXiv|first1=Emerging Technology from the|title=Mathematicians Solve Minimum Sudoku Problem|url=https://www.technologyreview.com/s/426554/mathematicians-solve-minimum-sudoku-problem/|access-date=27 November 2017|work=MIT Technology Review|language=en}}</ref>
*Symbolic validation (via [[computer algebra]]) of conjectures to motivate the search for an analytical proof
**Solutions to a special case of the quantum [[three-body problem]] known as the [[hydrogen molecule-ion]] were found standard quantum chemistry basis sets before realizing they all lead to the same unique analytical solution in terms of a ''generalization'' of the [[Lambert W function]].  Related to this work is the isolation of a previously unknown link between gravity theory and quantum mechanics in lower dimensions (see [[Quantum gravity#The dilaton|quantum gravity]] and references therein).
**In the realm of relativistic [[N-body problem|many-bodied mechanics]], namely the [[t-symmetry|time-symmetric]] [[Wheeler–Feynman absorber theory]]: the equivalence between an advanced [[Liénard–Wiechert potential]] of particle ''j'' acting on particle ''i'' and the corresponding potential for particle ''i'' acting on particle ''j'' was demonstrated exhaustively to order <math> 1/c^{10} </math> before being proved mathematically.  The Wheeler-Feynman theory has regained interest because of [[quantum nonlocality]].
**In the realm of linear optics, verification of the series expansion of the [[Slowly varying envelope approximation|envelope]] of the electric field for [[Ultrashort pulse#Wave packet propagation in nonisotropic media|ultrashort light pulses travelling in non isotropic media]].  Previous expansions had been incomplete: the outcome revealed an extra term vindicated by experiment.
*Evaluation of [[series (mathematics)|infinite series]], [[infinite product]]s and [[integral]]s (also see [[symbolic integration]]), typically by carrying out a high precision numerical calculation, and then using an [[integer relation algorithm]] (such as the [[Inverse Symbolic Calculator]]) to find a linear combination of mathematical constants that matches this value. For example, the following identity was rediscovered by Enrico Au-Yeung, a student of [[Jonathan Borwein]] using computer search and [[PSLQ algorithm]] in 1993:<ref>{{cite journal |author=Bailey, David |title=New Math Formulas Discovered With Supercomputers |journal=NAS News |year=1997 |volume=2 |issue=24 |url=https://www.nas.nasa.gov/About/Gridpoints/PDF/nasnews_V02_N24_1997.pdf}}</ref><ref>H. F. Sandham and Martin Kneser, The American mathematical monthly, Advanced problem 4305, Vol. 57, No. 4 (Apr., 1950), pp. 267-268</ref>

::<math>
\begin{align}
\sum_{k=1}^\infty \frac{1}{k^2}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{k}\right)^2 = \frac{17\pi^4}{360}.
\end{align}</math>

*Visual investigations
**In [[Indra's Pearls (book)|Indra's Pearls]], [[David Mumford]] and others investigated various properties of [[Möbius transformation]] and the [[Schottky group]] using computer generated images of the [[Group (mathematics)|groups]] which: ''furnished convincing evidence for many conjectures and lures to further exploration''.<ref>{{cite book | last = Mumford | first = David |author2=Series, Caroline |author3=Wright, David |title = Indra's Pearls: The Vision of Felix Klein | publisher = Cambridge | date = 2002 | isbn = 978-0-521-35253-6 |pages=viii}}</ref>