Experimental mathematics is an approach tomathematics in which computation is used to investigate mathematical objects and identify properties and patterns.^[1] It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration ofconjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit."^[2]

As expressed byPaul Halmos: "Mathematics is not adeductive science—that's a cliché. When you try to prove a theorem, you don't just list thehypotheses, and then start to reason. What you do istrial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does."^[3]

Mathematicians have always practiced experimental mathematics. Existing records of early mathematics, such asBabylonian mathematics, typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing results in a final, formal and abstract presentation. The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten.

Experimental mathematics as a separate area of study re-emerged in the twentieth century, when the invention of the electronic computer vastly increased the range of feasible calculations, with a speed and precision far greater than anything available to previous generations of mathematicians. A significant milestone and achievement of experimental mathematics was the discovery in 1995 of theBailey–Borwein–Plouffe formula for the binary digits ofπ. This formula was discovered not by formal reasoning, but insteadby numerical searches on a computer; only afterwards was a rigorousproof found.^[4]

The objectives of experimental mathematics are "to generate understanding and insight; to generate and confirm or confront conjectures; and generally to make mathematics more tangible, lively and fun for both the professional researcher and the novice".^[5]

The uses of experimental mathematics have been defined as follows:^[6]

1. Gaining insight and intuition.
2. Discovering new patterns and relationships.
3. Using graphical displays to suggest underlying mathematical principles.
4. Testing and especially falsifying conjectures.
5. Exploring a possible result to see if it is worth formal proof.
6. Suggesting approaches for formal proof.
7. Replacing lengthy hand derivations with computer-based derivations.
8. Confirming analytically derived results.

Experimental mathematics makes use ofnumerical methods to calculate approximate values forintegrals andinfinite series.Arbitrary precision arithmetic is often used to establish these values to a high degree of precision – typically 100 significant figures or more.Integer relation algorithms are then used to search for relations between these values andmathematical constants. Working with high precision values reduces the possibility of mistaking amathematical coincidence for a true relation. A formal proof of a conjectured relation will then be sought – it is often easier to find a formal proof once the form of a conjectured relation is known.

If acounterexample is being sought or a large-scaleproof by exhaustion is being attempted,distributed computing techniques may be used to divide the calculations between multiple computers.

Frequent use is made of generalmathematical software or domain-specific software written for attacks on problems that require high efficiency. Experimental mathematics software usually includeserror detection and correction mechanisms, integrity checks and redundant calculations designed to minimise the possibility of results being invalidated by a hardware or software error.

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 toEuler's sum of powers conjecture.
  • TheZetaGrid project was set up to search for a counterexample to theRiemann hypothesis.
  • Tomás Oliveira e Silva^[7] searched for a counterexample to theCollatz conjecture.
• Finding new examples of numbers or objects with particular properties
  • TheGreat Internet Mersenne Prime Search is searching for newMersenne primes.
  • The Great Periodic Path Hunt is searching for new periodic paths.
  • distributed.net's OGR project searched for optimalGolomb rulers.
  • ThePrimeGrid project is searching for the smallestRiesel andSierpiński numbers.
• Finding serendipitous numerical patterns
  • Edward Lorenz found theLorenz attractor, an early example of a chaoticdynamical system, by investigating anomalous behaviours in a numerical weather model.
  • TheUlam spiral was discovered by accident.
  • The pattern in theUlam numbers was discovered by accident.
  • Mitchell Feigenbaum's discovery of theFeigenbaum 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 acomputer-assistedproof by exhaustion
  • Thomas Hales's proof of theKepler conjecture.
  • Various proofs of thefour colour theorem.
  • Clement Lam's proof of the non-existence of afinite projective plane of order 10.^[8]
  • Gary McGuire proved a minimum uniquely solvableSudoku requires 17 clues.^[9]
• Symbolic validation (viacomputer algebra) of conjectures to motivate the search for an analytical proof
  • Solutions to a special case of the quantumthree-body problem known as thehydrogen molecule-ion were found standard quantum chemistry basis sets before realizing they all lead to the same unique analytical solution in terms of ageneralization of theLambert W function. Related to this work is the isolation of a previously unknown link between gravity theory and quantum mechanics in lower dimensions (seequantum gravity and references therein).
  • In the realm of relativisticmany-bodied mechanics, namely thetime-symmetricWheeler–Feynman absorber theory: the equivalence between an advancedLiénard–Wiechert potential of particlej acting on particlei and the corresponding potential for particlei acting on particlej was demonstrated exhaustively to order${\displaystyle 1/c^{10}}$ before being proved mathematically. The Wheeler-Feynman theory has regained interest because ofquantum nonlocality.
  • In the realm of linear optics, verification of the series expansion of theenvelope of the electric field forultrashort light pulses travelling in non isotropic media. Previous expansions had been incomplete: the outcome revealed an extra term vindicated by experiment.
• Evaluation ofinfinite series,infinite products andintegrals (also seesymbolic integration), typically by carrying out a high precision numerical calculation, and then using aninteger relation algorithm (such as theInverse 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 ofJonathan Borwein using computer search andPSLQ algorithm in 1993:^[10][11]

${\displaystyle \begin{array}{r}\sum _{k=1}^{\mathrm{\infty }}\frac{H_k^2}{k^2}=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2}{\left(\underset{H_k}{\underbrace{1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{k}}}\right)}^2=\frac{17{\pi }^4}{360}\end{array}}$

• Visual investigations
  • InIndra's Pearls,David Mumford and others investigated various properties ofMöbius transformation and theSchottky group using computer generated images of thegroups which:furnished convincing evidence for many conjectures and lures to further exploration.^[12]

Some plausible relations hold to a high degree of accuracy, but are still not true. One example is:

${\displaystyle {\int }_0^{\mathrm{\infty }}\cos (2x)\prod _{n=1}^{\mathrm{\infty }}\cos \left(\frac{x}{n}\right)\mathrm{d}x=\frac{\pi }{8}.}$

The two sides of this expression actually differ after the 42nd decimal place.^[13]

Another example is that the maximumheight (maximum absolute value of coefficients) of all the factors ofxⁿ − 1 appears to be the same as the height of thenthcyclotomic polynomial. This was shown by computer to be true forn < 10000 and was expected to be true for alln. However, a larger computer search showed that this equality fails to hold forn = 14235, when the height of thenth cyclotomic polynomial is 2, but maximum height of the factors is 3.^[14]

The followingmathematicians andcomputer scientists have made significant contributions to the field of experimental mathematics:

• Borwein integral
• Computer-aided proof
• Proofs and Refutations
• Experimental Mathematics (journal)
• Institute for Experimental Mathematics

• Experimental Mathematics (Journal)
• Centre for Experimental and Constructive Mathematics (CECM) atSimon Fraser University
• Collaborative Group for Research in Mathematics Education atUniversity of Southampton
• Recognizing Numerical Constants byDavid H. Bailey andSimon Plouffe
• Psychology of Experimental Mathematics
• Experimental Mathematics Website (Links and resources)
• The Great Periodic Path Hunt Website (Links and resources)
• An Algorithm for the Ages: PSLQ, A Better Way to Find Integer Relations (AlternativelinkArchived 2021-02-13 at theWayback Machine)
• Experimental Algorithmic Information Theory
• Sample Problems of Experimental Mathematics byDavid H. Bailey andJonathan M. Borwein
• Ten Problems in Experimental MathematicsArchived 2011-06-10 at theWayback Machine byDavid H. Bailey,Jonathan M. Borwein, Vishaal Kapoor, andEric W. Weisstein
• Institute for Experimental MathematicsArchived 2015-02-10 at theWayback Machine atUniversity of Duisburg-Essen

• Experimental mathematics
• Fields of mathematics

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