66:, 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.
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69:
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
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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
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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".
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659:{\displaystyle {\begin{aligned}\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{aligned}}}
156:
for a true relation. A formal proof of a conjectured relation will then be sought â it is often easier to find a formal proof once the form of a conjectured relation is known.
496:) 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
450:
182:
mechanisms, integrity checks and redundant calculations designed to minimise the possibility of results being invalidated by a hardware or software error.
54:, 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."
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or domain-specific software written for attacks on problems that require high efficiency. Experimental mathematics software usually includes
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379:. Related to this work is the isolation of a previously unknown link between gravity theory and quantum mechanics in lower dimensions (see
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78:. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous
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were found standard quantum chemistry basis sets before realizing they all lead to the same unique analytical solution in terms of a
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H. F. Sandham and Martin Kneser, The
American mathematical monthly, Advanced problem 4305, Vol. 57, No. 4 (Apr., 1950), pp. 267-268
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831:{\displaystyle \int _{0}^{\infty }\cos(2x)\prod _{n=1}^{\infty }\cos \left({\frac {x}{n}}\right)\mathrm {d} x={\frac {\pi }{8}}.}
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is often used to establish these values to a high degree of precision â typically 100 significant figures or more.
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Mathematicians have always practiced experimental mathematics. Existing records of early mathematics, such as
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Some plausible relations hold to a high degree of accuracy, but are still not true. One example is:
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significant milestone and achievement of experimental mathematics was the discovery in 1995 of the
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before being proved mathematically. The
Wheeler-Feynman theory has regained interest because of
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experimental (in either the
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Roger Frye used experimental mathematics techniques to find the smallest counterexample to
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and more informal beliefs and a careful analysis of the data acquired in this pursuit."
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furnished convincing evidence for many conjectures and lures to further exploration
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Use of computer programs to check a large but finite number of cases to complete a
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The two sides of this expression actually differ after the 42nd decimal place.
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have made significant contributions to the field of experimental mathematics:
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techniques may be used to divide the calculations between multiple computers.
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was based initially on numerical observations, followed by a rigorous proof.
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152:. Working with high precision values reduces the possibility of mistaking a
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In the realm of linear optics, verification of the series expansion of the
46:âthat's a clichĂ©. When you try to prove a theorem, you don't just list the
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An
Algorithm for the Ages: PSLQ, A Better Way to Find Integer Relations
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th cyclotomic polynomial is 2, but maximum height of the factors is 3.
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Using graphical displays to suggest underlying mathematical principles.
47:
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Finding new examples of numbers or objects with particular properties
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282:, by investigating anomalous behaviours in a numerical weather model.
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Replacing lengthy hand derivations with computer-based derivations.
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The uses of experimental mathematics have been defined as follows:
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Mathematics by
Experiment: Plausible Reasoning in the 21st Century
1117:
Mathematics by
Experiment: Plausible Reasoning in the 21st Century
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The Great
Periodic Path Hunt is searching for new periodic paths.
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360:) of conjectures to motivate the search for an analytical proof
848:(maximum absolute value of coefficients) of all the factors of
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Applications and examples of experimental mathematics include:
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are then used to search for relations between these values and
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Exploring a possible result to see if it is worth formal proof.
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Institute of
Electronics and Informatics Engineering of Aveiro
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TomĂĄs
Oliveira e Silva searched for a counterexample to the
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Collaborative Group for Research in Mathematics Education
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ultrashort light pulses travelling in non isotropic media
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project was set up to search for a counterexample to the
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Mumford, David; Series, Caroline; Wright, David (2002).
1190:"The Search for a Finite Projective Plane of Order 10"
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by J. Borwein, P. Borwein, R. Girgensohn and S. Parnes
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1167:"Computational verification of the 3x+1 conjecture"
16:For the mathematical journal of the same name, see
1316:Future Prospects for Computer-Assisted Mathematics
1257:"New Math Formulas Discovered With Supercomputers"
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658:
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1338:is 3 and 14235 = 3 x 4745. See Sloane sequences
1068:I Want to be a Mathematician: An Automathography
852:â 1 appears to be the same as the height of the
347:Gary McGuire proved a minimum uniquely solvable
864:< 10000 and was expected to be true for all
194:Searching for a counterexample to a conjecture
683:and others investigated various properties of
107:Testing and especially falsifying conjectures.
1231:"Mathematicians Solve Minimum Sudoku Problem"
409:and the corresponding potential for particle
8:
860:. This was shown by computer to be true for
1447:Sample Problems of Experimental Mathematics
1442:Experimental Algorithmic Information Theory
363:Solutions to a special case of the quantum
101:Discovering new patterns and relationships.
50:, and then start to reason. What you do is
1314:David H. Bailey and Jonathan M. Borwein,
1292:Indra's Pearls: The Vision of Felix Klein
1140:Borwein, Jonathan; Bailey, David (2004).
1115:Borwein, Jonathan; Bailey, David (2004).
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268:Finding serendipitous numerical patterns
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119:Confirming analytically derived results.
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691:using computer generated images of the
417:was demonstrated exhaustively to order
113:Suggesting approaches for formal proof.
1484:Institute for Experimental Mathematics
1408:Psychology of Experimental Mathematics
1049:Experimental Mathematics: A Discussion
1006:Institute for Experimental Mathematics
397:: the equivalence between an advanced
255:project is searching for the smallest
128:Experimental mathematics makes use of
1229:arXiv, Emerging Technology from the.
7:
1419:The Great Periodic Path Hunt Website
844:Another example is that the maximum
244:'s OGR project searched for optimal
229:Great Internet Mersenne Prime Search
132:to calculate approximate values for
1177:from the original on 18 March 2013.
340:'s proof of the non-existence of a
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18:Experimental Mathematics (journal)
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1165:Silva, TomĂĄs (28 December 2015).
163:is being sought or a large-scale
1413:Experimental Mathematics Website
1070:(1985), p. 321 (in 2013 reprint)
872:= 14235, when the height of the
278:, an early example of a chaotic
199:Euler's sum of powers conjecture
174:Frequent use is made of general
1395:Recognizing Numerical Constants
395:WheelerâFeynman absorber theory
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180:error detection and correction
142:Arbitrary precision arithmetic
98:Gaining insight and intuition.
72:BaileyâBorweinâPlouffe formula
1:
1195:American Mathematical Monthly
1119:. A.K. Peters. pp. vii.
386:In the realm of relativistic
1495:University of Duisburg-Essen
1294:. Cambridge. pp. viii.
706:Plausible but false examples
494:Inverse Symbolic Calculator
296:was discovered by accident.
289:was discovered by accident.
146:Integer relation algorithms
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1188:Clement W. H. Lam (1991).
1144:. A.K. Peters. p. 2.
1029:"Experimental Mathematics"
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500:using computer search and
490:integer relation algorithm
463:of the electric field for
399:LiĂ©nardâWiechert potential
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1390:University of Southampton
356:Symbolic validation (via
186:Applications and examples
74:for the binary digits of
1510:Experimental mathematics
1371:Experimental Mathematics
999:Experimental Mathematics
712:mathematical coincidence
445:{\displaystyle 1/c^{10}}
383:and references therein).
154:mathematical coincidence
42:: "Mathematics is not a
24:Experimental mathematics
1381:Simon Fraser University
342:finite projective plane
1477:, Vishaal Kapoor, and
1255:Bailey, David (1997).
992:Proofs and Refutations
928:Thomas Callister Hales
832:
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672:Visual investigations
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330:Various proofs of the
150:mathematical constants
64:Babylonian mathematics
1421:(Links and resources)
1415:(Links and resources)
1235:MIT Technology Review
858:cyclotomic polynomial
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685:Möbius transformation
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388:many-bodied mechanics
369:hydrogen molecule-ion
231:is searching for new
176:mathematical software
169:distributed computing
986:Computer-aided proof
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486:symbolic integration
421:
302:'s discovery of the
167:is being attempted,
124:Tools and techniques
1475:Jonathan M. Borwein
1455:Jonathan M. Borwein
1095:Jonathan M. Borwein
890:computer scientists
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454:quantum nonlocality
413:acting on particle
405:acting on particle
332:four colour theorem
316:proof by exhaustion
304:Feigenbaum constant
300:Mitchell Feigenbaum
292:The pattern in the
165:proof by exhaustion
86:Objectives and uses
1489:2015-02-10 at the
1465:2011-06-10 at the
1434:2021-02-13 at the
1321:2011-07-20 at the
1085:2011-09-27 at the
1054:2008-01-21 at the
1026:Weisstein, Eric W.
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377:Lambert W function
365:three-body problem
351:requires 17 clues.
217:Collatz conjecture
210:Riemann hypothesis
26:is an approach to
1479:Eric W. Weisstein
1301:978-0-521-35253-6
1151:978-1-56881-211-3
1126:978-1-56881-211-3
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1080:The Quest for Pi
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323:'s proof of the
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276:Lorenz attractor
38:As expressed by
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1019:
998:
990:
933:Donald Knuth
883:
873:
869:
865:
861:
853:
849:
843:
840:
715:
696:
414:
410:
406:
402:
401:of particle
372:
344:of order 10.
321:Thomas Hales
294:Ulam numbers
189:
173:
158:
127:
93:
89:
68:
61:
37:
23:
22:
1240:27 November
938:Clement Lam
338:Clement Lam
287:Ulam spiral
40:Paul Halmos
33:conjectures
28:mathematics
1012:References
484:(also see
274:found the
261:SierpiĆski
48:hypotheses
1373:(Journal)
1034:MathWorld
1001:(journal)
818:π
784:
776:∞
761:∏
745:
737:∞
728:∫
635:π
598:⋯
538:∞
523:∑
482:integrals
253:PrimeGrid
134:integrals
1504:Category
1487:Archived
1463:Archived
1432:Archived
1319:Archived
1264:NAS News
1175:Archived
1083:Archived
1052:Archived
975:See also
687:and the
504:in 1993:
461:envelope
263:numbers.
206:ZetaGrid
1355:A160338
1353::
1345:A137979
1343::
1216:2323798
695:which:
375:of the
82:found.
58:History
1298:
1214:
1148:
1123:
846:height
693:groups
349:Sudoku
257:Riesel
1270:(24).
1260:(PDF)
1212:JSTOR
159:If a
80:proof
1453:and
1429:link
1401:and
1351:OEIS
1348:and
1341:OEIS
1336:4745
1296:ISBN
1242:2017
1146:ISBN
1121:ISBN
1101:and
888:and
480:and
285:The
259:and
251:The
227:The
204:The
136:and
1493:at
1469:by
1449:by
1397:by
1388:at
1379:at
1204:doi
1089:by
856:th
781:cos
742:cos
675:In
645:360
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1266:.
1262:.
1233:.
1210:.
1200:98
1198:.
1192:.
1173:.
1169:.
1097:,
1093:,
1031:.
679:,
631:17
476:,
438:10
140:.
1438:)
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1268:2
1244:.
1218:.
1206::
1154:.
1129:.
1105:.
1037:.
874:n
870:n
866:n
862:n
854:n
850:x
826:.
821:8
813:=
810:x
806:d
801:)
796:n
793:x
788:(
771:1
768:=
765:n
757:)
754:x
751:2
748:(
732:0
699:.
650:.
639:4
625:=
620:2
615:)
609:k
606:1
601:+
595:+
590:3
587:1
582:+
577:2
574:1
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566:1
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549:k
545:1
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530:=
527:k
456:.
434:c
429:/
425:1
415:j
411:i
407:i
403:j
334:.
327:.
248:.
235:.
219:.
212:.
201:.
76:Ï
20:.
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