A natural partial order on the prime numbers

Lucian M. Ionescu
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 1, Pages 1—9
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Authors and affiliations

Lucian M. Ionescu
Department of Mathematics, Illinois State University
IL 61790-4520, United States

Abstract

A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields [1], independently of [2], who investigated Pratt trees  [3] used for primality tests. It leads to a correspondence with the Hopf algebra of rooted trees, and as an application, to an alternative approach to the Prime Number Theorem.

Keywords

  • Prime numbers
  • Pratt tress
  • Rooted trees
  • Prime Number Theorem
  • Finite fields

AMS Classification

  • 11NXX
  • 11TXX

References

  1. Ionescu, L. M. (2011) Prime Numbers and Multiplicative Number Theory, Discrete Mathematics Seminar, 10/2011,
    http://my.ilstu.edu/~lmiones/presentations_drafts.htm
  2. Ford, K., S. V. Konyagin, & F. Luca. (2010) Prime chains and Pratt trees, 0904.0473.
  3. Pratt, V. R. (1975) Every prime has a succint certificat, SIAM J. Comput., 4(3), 214–220.
  4. Manin, Y. Lectures on zeta functions and motives, MPI / 92–50.
  5. Shai Haran, M. J. (2001) The mystery of the real prime.
  6. Tretkoff, P. (2006) Noncommutative geometry and number theory, Clay Mathematics Proceedings, Vol. 6.
  7. Ionescu, L. M. (2011) Remarks on physics as number theory, 2011, http://www.gsjournal.net/old/files/4606_Ionescu2.pdf
  8. Ionescu, L. M. (2004) From Lie theory to deformation theory, http://arxiv.org/abs/0704.2213.
  9. Ionescu, L. M. (2013–2014) p-adic math-physics, Outlines for presentations in the Math-Physics seminar.
  10. Ionescu, L. M. p-adic numbers and algebraic quantum groups. (work in progress)
  11. Ionescu, L. M. The quantum group of rationals,
    http://my.ilstu.edu/~lmiones/presentations_drafts.htm
  12. Formal groups. Wikipedia, The Free Encyclopedia.
  13. Ionescu, L. M. Real numbers and p-adic numbers: a Haar analysis point of view. (work in progress)
  14. Ionescu, L. M. (2012, 2013) ISU Summer Research Academy, http://cemast2012.webs.com/prime-numbers.
  15. Gracia-Bondia, J. M., Varilly, J. C., & Figueroa, H. (2001) Elements of Noncommutative Geometry, Birkhauser Basel.

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Cite this paper

APA

Ionescu, L. M. (2015). A natural partial order on the prime numbers. Notes on Number Theory and Discrete Mathematics, 20(1), 1-9.

Chicago

Ionescu, Lucian M. “A Natural Partial Order on the Prime Numbers.” Notes on Number Theory and Discrete Mathematics 21, no. 1 (2015): 1-9.

MLA

Ionescu, Lucian M. “A Natural Partial Order on the Prime Numbers.” Notes on Number Theory and Discrete Mathematics 21.1 (2015): 1-9. Print.

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