Article

Predicting prime numbers using cartesian genetic programming

Authors:

James Alfred Walker,

Julian Francis MillerAuthors Info & Claims

EuroGP'07: Proceedings of the 10th European conference on Genetic programming

Pages 205 - 216

Published: 11 April 2007 Publication History

Abstract

Prime generating polynomial functions are known that can produce sequences of prime numbers (e.g. Euler polynomials). However, polynomials which produce consecutive prime numbers are much more difficult to obtain. In this paper, we propose approaches for both these problems. The first uses Cartesian Genetic Programming (CGP) to directly evolve integer based prime-prediction mathematical formulae. The second uses multi-chromosome CGP to evolve a digital circuit, which represents a polynomial. We evolved polynomials that can generate 43 primes in a row. We also found functions capable of producing the first 40 consecutive prime numbers, and a number of digital circuits capable of predicting up to 208 consecutive prime numbers, given consecutive input values. Many of the formulae have been previously unknown.

References

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Wells, D.: Prime Numbers. John Wiley and sons (2005).

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Euler, L.: Extrait d'un lettre de m. euler le pere a m. bernoulli concernant le memoire imprime parmi ceux de 1771. Nouveaux Mémoires de l'Académie royale des Sciences de Berlin, Histoire (1772) 35-36.

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Legendre, A.M.: Théorie des nombres. 2 edn. Libraire Scientifique A. Herman (1808).

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Mollin, R.: Quadratics. Boca Raton (1995).

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Mollin, R.: Prime-producing quadratics. American Mathematical Monthly 104(6) (1997) 529-544.

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Pegg Jnr., E.: Math games: Prime generating polynomials.

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Fung, G., Williams, H.: Quadratic polynomials which have a high density of prime values. Mathematics of Computation 55 (1990) 345-353.

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Mollin, R.: New prime-producing quadratic polynomials associated with class number one or two. New York Journal of Mathematics 8 (2002) 161-168.

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Harrell, H.: Prime Producing Equations: The Distribution of Primes and Composites Within a Special Number Arrangement. AuthorHouse (2002).

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Miller, J.F., Thomson, P.: Cartesian genetic programming. In: Proceedings of the 3rd European Conference on Genetic Programming (EuroGP 2000). Volume 1802 of LNCS., Edinburgh, UK, Springer-Verlag (15-16 April 2000) 121-132.

Digital Library

[11]

Walker, J.A., Miller, J.F.: A multi-chromosome approach to standard and embedded cartesian genetic programming. In: Proceedings of the 2006 Genetic and Evolutionary Computation Conference (GECCO 2006). Volume 1., Seattle, Washington, USA, ACM Press (8-12 July 2006) 903-910.

Digital Library

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Koza, J.R.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press (1992).

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Poli, R.: Parallel Distributed Genetic Programming. Technical Report CSRP-96- 15, School of Computer Science, University of Birmingham, B15 2TT, UK (September 1996).

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Yu, T., Miller, J.F.: Neutrality and the evolvability of boolean function landscape. In: Proceedings of the 4th European Conference on Genetic Programming (EuroGP 2001). Volume 2038 of Lecture Notes in Computer Science., Springer-Verlag (2001) 204-217.

Digital Library

[15]

Vassilev, V.K., Miller, J.F.: The advantages of landscape neutrality in digital circuit evolution. In: Proceedings of the 3rd International Conference on Evolvable Systems (ICES 2000). Volume 1801 of Lecture Notes in Computer Science., Springer Verlag (2000) 252-263.

Digital Library

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Walker, J.A., Miller, J.F.: The automatic acquisition, evolution and re-use of modules in cartesian genetic programming. Accepted for publication in IEEE Transactions on Evolutionary Computation.

Cited By

Khan MMendes AZhang PChalup S(2017)Evolving multi-dimensional wavelet neural networks for classification using Cartesian Genetic ProgrammingNeurocomputing10.1016/j.neucom.2017.03.048247:C(39-58)Online publication date: 19-Jul-2017
https://dl.acm.org/doi/10.1016/j.neucom.2017.03.048
Khan MChalup SMendes A(2016)Parkinson's Disease Data Classification Using Evolvable Wavelet Neural NetworksProceedings of the Second Australasian Conference on Artificial Life and Computational Intelligence - Volume 959210.1007/978-3-319-28270-1_10(113-124)Online publication date: 2-Feb-2016
https://dl.acm.org/doi/10.1007/978-3-319-28270-1_10
Miller JTurner AEsparcia-Alcázar A(2015)Cartesian Genetic ProgrammingProceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation10.1145/2739482.2756571(179-198)Online publication date: 11-Jul-2015
https://dl.acm.org/doi/10.1145/2739482.2756571
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Index Terms

Predicting prime numbers using cartesian genetic programming
1. Computing methodologies
  1. Artificial intelligence
    1. Search methodologies
      1. Heuristic function construction

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Guide Proceedings

EuroGP'07: Proceedings of the 10th European conference on Genetic programming

April 2007

382 pages

ISBN:9783540716020

Editors:
Marc Ebner
Universität Würzburg, Germany
,
Michael O'Neill
University College Dublin, Ireland
,
Anikó Ekárt
Aston University, Birmingham, UK
,
Leonardo Vanneschi
University of Milano-Bicocca, Italy
,
Anna Isabel Esparcia-Alcázar
Universidad Politécnica de Valencia, Spain

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 11 April 2007

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Cited By

Khan MMendes AZhang PChalup S(2017)Evolving multi-dimensional wavelet neural networks for classification using Cartesian Genetic ProgrammingNeurocomputing10.1016/j.neucom.2017.03.048247:C(39-58)Online publication date: 19-Jul-2017
https://dl.acm.org/doi/10.1016/j.neucom.2017.03.048
Khan MChalup SMendes A(2016)Parkinson's Disease Data Classification Using Evolvable Wavelet Neural NetworksProceedings of the Second Australasian Conference on Artificial Life and Computational Intelligence - Volume 959210.1007/978-3-319-28270-1_10(113-124)Online publication date: 2-Feb-2016
https://dl.acm.org/doi/10.1007/978-3-319-28270-1_10
Miller JTurner AEsparcia-Alcázar A(2015)Cartesian Genetic ProgrammingProceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation10.1145/2739482.2756571(179-198)Online publication date: 11-Jul-2015
https://dl.acm.org/doi/10.1145/2739482.2756571
Miller JAlba E(2013)GECCO 2013 tutorialProceedings of the 15th annual conference companion on Genetic and evolutionary computation10.1145/2464576.2464578(715-740)Online publication date: 6-Jul-2013
https://dl.acm.org/doi/10.1145/2464576.2464578
Miller JHarding SMoore J(2012)GECCO 2012 tutorialProceedings of the 14th annual conference companion on Genetic and evolutionary computation10.1145/2330784.2330932(1093-1116)Online publication date: 7-Jul-2012
https://dl.acm.org/doi/10.1145/2330784.2330932
McDermott JWhite DLuke SManzoni LCastelli MVanneschi LJaskowski WKrawiec KHarper RDe Jong KO'Reilly UMoore J(2012)Genetic programming needs better benchmarksProceedings of the 14th annual conference on Genetic and evolutionary computation10.1145/2330163.2330273(791-798)Online publication date: 7-Jul-2012
https://dl.acm.org/doi/10.1145/2330163.2330273
Miller JHarding SLanzi P(2011)GECCO 2011 tutorialProceedings of the 13th annual conference companion on Genetic and evolutionary computation10.1145/2001858.2002136(1261-1284)Online publication date: 12-Jul-2011
https://dl.acm.org/doi/10.1145/2001858.2002136
Cazenave T(2010)Nested Monte-Carlo Expression DiscoveryProceedings of the 2010 conference on ECAI 2010: 19th European Conference on Artificial Intelligence10.5555/1860967.1861203(1057-1058)Online publication date: 4-Aug-2010
https://dl.acm.org/doi/10.5555/1860967.1861203
Miller JHarding SPelikan MBranke J(2010)Cartesian genetic programmingProceedings of the 12th annual conference companion on Genetic and evolutionary computation10.1145/1830761.1830924(2927-2948)Online publication date: 7-Jul-2010
https://dl.acm.org/doi/10.1145/1830761.1830924
Trefzer MKuyucu TMiller JTyrrell A(2009)A model for intrinsic artificial development featuring structural feedback and emergent growthProceedings of the Eleventh conference on Congress on Evolutionary Computation10.5555/1689599.1689639(301-308)Online publication date: 18-May-2009
https://dl.acm.org/doi/10.5555/1689599.1689639
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