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Random walk on the prime numbers

Author

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  • Wolf, Marek

Abstract

The one-dimensional random walk (RW), where steps up and down are performed according to the occurrence of special primes, is defined. Some quantities characterizing RW are investigated. The mean fluctuation function F(l) displays perfect power-law dependence F(l)∼l1/2 indicating that the defined RW is not correlated. The number of returns of this special RW to the origin is investigated. It turns out that this single, very special, realization of RW is a typical one in the sense that the usual characteristics used to measure RW, take values close to the ones averaged over all random walks. This fact suggests that random numbers of good quality could be obtained by means of RW on prime numbers. The fractal structure on the subset of primes is also found.

Suggested Citation

  • Wolf, Marek, 1998. "Random walk on the prime numbers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 250(1), pages 335-344.
    • Handle: RePEc:eee:phsmap:v:250:y:1998:i:1:p:335-344
    DOI: 10.1016/S0378-4371(97)00661-4
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    Citations

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    Cited by:

    1. Gadiyar, H.Gopalkrishna & Padma, R., 1999. "Ramanujan–Fourier series, the Wiener–Khintchine formula and the distribution of prime pairs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(2), pages 503-510.
    2. Szpiro, George G, 2004. "The gaps between the gaps: some patterns in the prime number sequence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 341(C), pages 607-617.
    3. 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.
    4. 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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