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| An '''integer relation''' between a set of real numbers ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> is a set of integers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>n</sub>, not all 0, such that
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| :<math>a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\,</math>
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| An '''integer relation algorithm''' is an [[algorithm]] for finding integer relations. Specifically, given a set of real numbers known to a given precision, an integer relation algorithm will either find an integer relation between them, or will determine that no integer relation exists with coefficients whose magnitudes are less than a certain [[upper bound]].<ref>Since the set of real numbers can only be specified up to a finite precision, an algorithm that did not place limits on the size of its coefficients would always find an integer relation for sufficiently large coefficients. Results of interest occur when the size of the coefficients in an integer relation is small compared to the precision with which the real numbers are specified.</ref>
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| == History ==
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| For the case ''n'' = 2, an extension of the [[Euclidean algorithm]] can determine whether there is an integer relation between any two real numbers ''x''<sub>1</sub> and ''x''<sub>2</sub>. The algorithm generates successive terms of the [[continued fraction]] expansion of ''x''<sub>1</sub>/''x''<sub>2</sub>; if there is an integer relation between the numbers then their ratio is rational and the algorithm eventually terminates.
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| *The Ferguson–Forcade algorithm was published in 1979 by [[Helaman Ferguson]] and [[R.W. Forcade]].<ref>{{MathWorld|urlname=IntegerRelation|title=Integer Relation}}</ref> Although the paper treats general ''n'', it is not clear if the paper fully solves the problem because it lacks detailed steps, proofs, and a precision bound; crucial for a reliable implementation.
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| *The first algorithm with complete proofs was the '''[[Lenstra–Lenstra–Lovász lattice basis reduction algorithm|LLL algorithm]]''', developed by [[Arjen Lenstra]], [[Hendrik Lenstra]] and [[László Lovász]] in 1982.<ref>{{MathWorld|urlname=LLLAlgorithm|title=LLL Algorithm}}</ref>
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| *The '''HJLS algorithm''', developed by Johan Håstad, Bettina Just, Jeffrey Lagarias, and Claus-Peter Schnorr in 1986.<ref>{{MathWorld|urlname=HJLSAlgorithm|title=HJLS Algorithm}}</ref><ref>Johan Håstad, Bettina Just, Jeffrey Lagarias, Claus-Peter Schnorr: ''Polynomial time algorithms for finding integer relations among real numbers.'' Preliminary version: STACS 1986 (''Symposium Theoret. Aspects Computer Science'') Lecture Notes Computer Science 210 (1986), p. 105–118. ''SIAM J. Computing'', Vol. 18 (1989), p. 859–881</ref>
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| *The '''PSOS algorithm''', developed by Ferguson in 1988.<ref>{{MathWorld|urlname=PSOSAlgorithm|title=PSOS Algorithm}}</ref>
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| *The '''PSLQ algorithm''', developed by Ferguson and Bailey in 1992 and substantially simplified by Ferguson, Bailey, and Arno in 1999.<ref>{{MathWorld|urlname=PSLQAlgorithm|title=PSLQ Algorithm}}</ref><ref>[http://crd.lbl.gov/~dhbailey/dhbpapers/pslq.pdf ''A Polynomial Time, Numerically Stable Integer Relation Algorithm''] by Helaman R. P. Ferguson and David H. Bailey; RNR Technical Report RNR-91-032; July 14, 1992</ref> In 2000 the PSLQ algorithm was selected as one of the "Top Ten Algorithms of the Century" by [[Jack Dongarra]] and Francis Sullivan<ref>{{cite journal |author=Barry A. Cipra |url=http://www.uta.edu/faculty/rcli/TopTen/topten.pdf |title=The Best of the 20th Century: Editors Name Top 10 Algorithms |journal=SIAM News |volume=33 |issue=4}}</ref> even though it is essentially equivalent to HJLS.<ref>Jingwei Chen, Damien Stehlé, Gilles Villard: ''[http://perso.ens-lyon.fr/damien.stehle/downloads/PSLQHJLS.pdf A New View on HJLS and PSLQ: Sums and Projections of Lattices.''], [http://www.issac-conference.org/2013/ ISSAC'13]</ref>
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| *The LLL algorithm has been improved by numerous authors. Modern implementations can solve integer relation problems with ''n'' well above 500.
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| ==Applications==
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| Integer relation algorithms have numerous applications. The first application is to determine whether a given real number ''x'' is likely to be [[algebraic number|algebraic]], by searching for an integer relation between a set of powers of ''x'' {1, ''x'', ''x''<sup>2</sup>, ..., ''x''<sup>''n''</sup>}. The second application is to search for an integer relation between a real number ''x'' and a set of mathematical constants such as ''e'', π and ln(2), which will lead to an expression for ''x'' as a linear combination of these constants.
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| A typical approach in [[experimental mathematics]] is to use [[numerical method]]s and [[arbitrary precision arithmetic]] to find an approximate value for an [[series (mathematics)|infinite series]], [[infinite product]] or an [[integral]] to a high degree of precision (usually at least 100 significant figures), and then use an integer relation algorithm to search for an integer relation between this value and a set of mathematical constants. If an integer relation is found, this suggests a possible [[closed-form expression]] for the original series, product or integral. This conjecture can then be validated by formal algebraic methods. The higher the precision to which the inputs to the algorithm are known, the greater the level of confidence that any integer relation that is found is not just a [[Almost integer|numerical artifact]].
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| A notable success of this approach was the use of the PSLQ algorithm to find the integer relation that led to the [[Bailey-Borwein-Plouffe formula]] for the value of [[pi|π]]. PSLQ has also helped find new identities involving [[multiple zeta function]]s and their appearance in [[quantum field theory]]; and in identifying bifurcation points of the [[logistic map]]. For example, where B<sub>4</sub> is the logistic map's fourth bifurcation point, the constant α=-B<sub>4</sub>(B<sub>4</sub>-2) appears to be a root of a 120th-degree polynomial whose largest coefficient is 257<sup>30</sup> or about 2·10<sup>72</sup>.<ref>David H. Bailey and David J. Broadhurst, [http://crd.lbl.gov/~dhbailey/dhbpapers/ppslq.pdf "Parallel Integer Relation Detection: Techniques and Applications,"] Mathematics of Computation, vol. 70, no. 236 (Oct 2000), pg. 1719-1736; LBNL-44481.</ref>
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| Integer relation algorithms are combined with tables of high precision mathematical constants and heuristic search methods in applications such as the [[Inverse Symbolic Calculator]] or [[Plouffe's Inverter]].
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| Integer relation finding can be used to factor polynomials over the rationals of high degree.<ref>M. van Hoeij: ''Factoring polynomials and the knapsack problem.'' J. of Number Theory, 95, 167-189, (2002).</ref>
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| == References ==
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| {{reflist|colwidth=30em}}
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| == External links ==
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| * [http://oldweb.cecm.sfu.ca/organics/papers/bailey/paper/html/paper.html ''Recognizing Numerical Constants''] by [[David H. Bailey]] and [[Simon Plouffe]]
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| * [http://crd.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf ''Ten Problems in Experimental Mathematics''] by David H. Bailey, [[Jonathan Borwein|Jonathan M. Borwein]], Vishaal Kapoor, and [[Eric W. Weisstein]]
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| {{number theoretic algorithms}}
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| [[Category:Number theoretic algorithms]]
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