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


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.


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

AMS Classification

  • 11NXX
  • 11TXX


  1. Ionescu, L. M. (2011) Prime Numbers and Multiplicative Number Theory, Discrete Mathematics Seminar, 10/2011,
  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.
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  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)
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  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

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

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