Experimental mathematics is an approach to

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

in which computation is used to investigate mathematical objects and identify properties and patterns. 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 of

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

s and more informal beliefs and a careful analysis of the data acquired in this pursuit." As expressed by Paul Halmos: "Mathematics is not a

deductive science Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how formal system, logical systems can be used to construct Validity (logic), valid and soundness, sound arguments, metalogic studies the properties of logical systems.Har ...

—that's a cliché. When you try to prove a theorem, you don't just list the

hypotheses A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obser ...

, and then start to reason. What you do is trial 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."

History

Mathematicians have always practiced experimental mathematics. Existing records of early mathematics, such as Babylonian 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 the

Bailey–Borwein–Plouffe formula The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for . It was discovered in 1995 by Simon Plouffe and is named after the authors of the article in which it was published, David H. Bailey, Peter Borwein, and Plouffe. Before that, ...

for the binary digits of π. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...

found.

Objectives and uses

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". The uses of experimental mathematics have been defined as follows: #Gaining insight and intuition. #Discovering new patterns and relationships. #Using graphical displays to suggest underlying mathematical principles. #Testing and especially falsifying conjectures. #Exploring a possible result to see if it is worth formal proof. #Suggesting approaches for formal proof. #Replacing lengthy hand derivations with computer-based derivations. #Confirming analytically derived results.

Tools and techniques

Experimental mathematics makes use of

numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

to calculate approximate values for

integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

s and infinite series.

Arbitrary precision arithmetic In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are li ...

is often used to establish these values to a high degree of precision – typically 100 significant figures or more.

Integer relation algorithm An integer relation between a set of real numbers ''x''1, ''x''2, ..., ''x'n'' is a set of integers ''a''1, ''a''2, ..., ''a'n'', not all 0, such that :a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\, An integer relation algorithm is an algorithm fo ...

s are then used to search for relations between these values and

mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...

s. Working with high precision values reduces the possibility of mistaking a mathematical 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 a counterexample is being sought or a large-scale proof by exhaustion is being attempted,

distributed computing A distributed system is a system whose components are located on different computer network, networked computers, which communicate and coordinate their actions by message passing, passing messages to one another from any system. Distributed com ...

techniques may be used to divide the calculations between multiple computers. Frequent use is made of general mathematical software or domain-specific software written for attacks on problems that require high efficiency. Experimental mathematics software usually includes

error detection and correction In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communi ...

mechanisms, integrity checks and redundant calculations designed to minimise the possibility of results being invalidated by a hardware or software error.

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 Euler's conjecture is a disproved conjecture in mathematics related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers is ...

. **The

ZetaGrid ZetaGrid was at one time the largest distributed computing project, designed to explore the non-trivial roots of the Riemann zeta function, checking over one billion roots a day. Roots of the zeta function are of particular interest in mathematic ...

project was set up to search for a counterexample to the

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...

. **Tomás Oliveira e Silva searched for a counterexample to the

Collatz conjecture The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...

. *Finding new examples of numbers or objects with particular properties **The

Great Internet Mersenne Prime Search The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and ...

is searching for new

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...

s. **The Great Periodic Path Hunt is searching for new periodic paths. **The

distributed.net Distributed.net is a volunteer computing effort that is attempting to solve large scale problems using otherwise idle CPU or GPU time. It is governed by Distributed Computing Technologies, Incorporated (DCTI), a non-profit organization under U. ...

's OGR project is searching for optimal

Golomb ruler In mathematics, a Golomb ruler is a set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its ''order'', and the largest distance between two of its mar ...

s. **The

Riesel Sieve Riesel Sieve was a volunteer computing project, running in part on the BOINC platform. Its aim was to prove that 509,203 is the smallest Riesel number, by finding a prime of the form for all odd smaller than 509,203. Progress At the start of th ...

project is searching for the smallest

Riesel number In mathematics, a Riesel number is an odd natural number ''k'' for which k\times2^n-1 is composite for all natural numbers ''n'' . In other words, when ''k'' is a Riesel number, all members of the following set are composite: :\left\. If the for ...

. **The

Seventeen or Bust Seventeen or Bust was a volunteer computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem. The project solved eleven cases before a server loss in April 2016 forced it to cease operations. Work on the ...

project is searching for the smallest Sierpinski number. *Finding serendipitous numerical patterns ** Edward Lorenz found the

Lorenz attractor The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lo ...

, an early example of a chaotic

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...

, by investigating anomalous behaviours in a numerical weather model. **The Ulam spiral was discovered by accident. **The pattern in the

Ulam number In mathematics, the Ulam numbers comprise an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with ''U''1 = 1 and ''U''2 =&nbs ...

s was discovered by accident. ** Mitchell Feigenbaum's discovery of the

Feigenbaum constant In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. History ...

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 Computer-aided or computer-assisted is an adjectival phrase that hints of the use of a computer as an indispensable tool in a certain field, usually derived from more traditional fields of science and engineering. Instead of the phrase computer-ai ...

proof by exhaustion ** 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

finite projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

of order 10. **Gary McGuire proved a minimum uniquely solvable Sudoku requires 17 clues. *Symbolic validation (via

computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...

) 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 The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . It consists of two hydrogen nuclei ( protons) sharing a single electron. It is the simplest molecular ion. The ion can be formed from the ionization of a ...

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 Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

and references therein). **In the realm of relativistic many-bodied mechanics, namely the time-symmetric Wheeler–Feynman absorber theory: the equivalence between an advanced

Liénard–Wiechert potential The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these descr ...

of particle ''j'' acting on particle ''i'' and the corresponding potential for particle ''i'' acting on particle ''j'' was demonstrated exhaustively to order

1/c^

before being proved mathematically. The Wheeler-Feynman theory has regained interest because of

quantum nonlocality In theoretical physics, quantum nonlocality refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not admit an interpretation in terms of a local realistic theory. Quantum nonlocality has been experimen ...

. **In the realm of linear optics, verification of the series expansion of the

envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...

of the electric field for ultrashort light pulses travelling in non isotropic media. Previous expansions had been incomplete: the outcome revealed an extra term vindicated by experiment. *Evaluation of infinite series, infinite products and

s (also see

symbolic integration In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or ''indefinite integral'', of a given function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that :\frac = f(x). This is also ...

), typically by carrying out a high precision numerical calculation, and then using an

integer relation algorithm An integer relation between a set of real numbers ''x''1, ''x''2, ..., ''x'n'' is a set of integers ''a''1, ''a''2, ..., ''a'n'', not all 0, such that :a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\, An integer relation algorithm is an algorithm fo ...

(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 Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they ...

using computer search and

PSLQ algorithm An integer relation between a set of real numbers ''x''1, ''x''2, ..., ''x'n'' is a set of integers ''a''1, ''a''2, ..., ''a'n'', not all 0, such that :a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\, An integer relation algorithm is an algorithm fo ...

in 1993: ::

\begin
\sum_^\infty \frac\left(1+\frac+\frac+\cdots+\frac\right)^2 = \frac.
\end

*Visual investigations **In Indra's Pearls, David Mumford and others investigated various properties of

Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...

and the

Schottky group In mathematics, a Schottky group is a special sort of Kleinian group, first studied by . Definition Fix some point ''p'' on the Riemann sphere. Each Jordan curve not passing through ''p'' divides the Riemann sphere into two pieces, and we call ...

using computer generated images of the

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

which: ''furnished convincing evidence for many conjectures and lures to further exploration''.

Plausible but false examples

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

\int_^\cos(2x)\prod_^\cos\left(\frac\right)\mathrmx = \frac.

The two sides of this expression actually differ after the 42nd decimal place.David H. Bailey and Jonathan M. Borwein
Future Prospects for Computer-Assisted Mathematics
December 2005 Another example is that the maximum

height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...

(maximum absolute value of coefficients) of all the factors of ''x''^''n'' − 1 appears to be the same as the height of the ''n''th

cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primiti ...

. This was shown by computer to be true for ''n'' < 10000 and was expected to be true for all ''n''. However, a larger computer search showed that this equality fails to hold for ''n'' = 14235, when the height of the ''n''th cyclotomic polynomial is 2, but maximum height of the factors is 3.The height of Φ₄₇₄₅ is 3 and 14235 = 3 x 4745. See Sloane sequences and .

Practitioners

The following

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

s and

computer scientist A computer scientist is a person who is trained in the academic study of computer science. Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus (al ...

s have made significant contributions to the field of experimental mathematics: *

Fabrice Bellard Fabrice Bellard (; born 1972) is a French computer programmer known for writing FFmpeg, QEMU, and the Tiny C Compiler. He developed Bellard's formula for calculating single digits of pi. In 2012, Bellard co-founded Amarisoft, a telecommunication ...

* David H. Bailey *

* David Epstein * Helaman Ferguson * Ronald Graham * Thomas Callister Hales * Donald Knuth * Clement Lam *

Oren Patashnik Oren Patashnik (born 1954) is an American computer scientist. He is notable for co-creating BibTeX, and co-writing '' Concrete Mathematics: A Foundation for Computer Science''. He is a researcher at the Center for Communications Research, La Jol ...

* Simon Plouffe *

Eric Weisstein Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and Encyclopedia, encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the ''CRC Concise Ency ...

* Stephen Wolfram * Doron Zeilberger * A.J. Han Vinck

References

{{reflist, 2

External links

Experimental Mathematics
(Journal)
Centre for Experimental and Constructive Mathematics (CECM)
at Simon Fraser University
Collaborative Group for Research in Mathematics Education
at

University of Southampton , mottoeng = The Heights Yield to Endeavour , type = Public research university , established = 1862 – Hartley Institution1902 – Hartley University College1913 – Southampton University Coll ...

Recognizing Numerical Constants
by David H. Bailey and Simon 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
(Alternativ

Experimental Algorithmic Information Theory

Sample Problems of Experimental Mathematics
by David H. Bailey and

Jonathan M. Borwein Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they ...

Ten Problems in Experimental Mathematics
by David H. Bailey,

, Vishaal Kapoor, and

Eric W. Weisstein Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...

Institute for Experimental Mathematics
at

University of Duisburg-Essen The University of Duisburg-Essen (german: link=no, Universität Duisburg-Essen) is a public research university in North Rhine-Westphalia, Germany. In the 2019 ''Times Higher Education World University Rankings'', the university was awarded ...