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The distribution of prime numbers: The solution comes from dynamical processes and genetic algorithms

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  • Iovane, Gerardo

Abstract

In this work, we show that the set of primes can be obtained through dynamical processes. Indeed, we see that behind their generation there is an apparent stochastic process; this is obtained with the combination of two processes: a “zig-zag” between two classes of primes and an intermittent process (that is a selection rule to exclude some prime candidates of the classes). Although we start with a stochastic process, the knowledge of its inner properties in terms of zig-zagging and intermittent processes gives us a deterministic and analytic way to generate the distribution of prime numbers. Thanks to genetic algorithms and evolution systems, as we will see, we answer some of most relevant questions of the last two centuries, that is “How can we know a priori if a number is prime or not? Or similarly, does the generation of number primes follow a specific rule and if yes what is its form? Moreover, has it a deterministic or stochastic form?” To reach these results we start to analyze prime numbers by using binary representation and building a hierarchy among derivative classes.

Suggested Citation

  • Iovane, Gerardo, 2008. "The distribution of prime numbers: The solution comes from dynamical processes and genetic algorithms," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 23-42.
    • Handle: RePEc:eee:chsofr:v:37:y:2008:i:1:p:23-42
    DOI: 10.1016/j.chaos.2007.10.017
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    1. Chen, Qingjiang & Liu, Baocang & Cao, Huaixin, 2009. "Construction of a sort of multiple pseudoframes for subspaces with filter banks," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 801-808.
    2. Iovane, Gerardo, 2009. "The set of prime numbers: Multiscale analysis and numeric accelerators," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1953-1965.
    3. Iovane, Gerardo, 2009. "The set of primes: Towards an optimized algorithm, prime generation and validation, and asymptotic consequences," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1344-1352.
    4. García-Sandoval, J.P., 2020. "Fractals and discrete dynamics associated to prime numbers," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    5. Cecen, Songul & Demirer, R. Murat & Bayrak, Coskun, 2009. "A new hybrid nonlinear congruential number generator based on higher functional power of logistic maps," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 847-853.
    6. Iovane, Gerardo, 2008. "The set of prime numbers: Symmetries and supersymmetries of selection rules and asymptotic behaviours," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 950-961.
    7. Iovane, Gerardo, 2009. "The set of prime numbers: Multifractals and multiscale analysis," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 1945-1958.

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