Research Article | | Peer-Reviewed

### Algebraic Representation of Primes by Hybrid Factorization

Received: 4 February 2024     Accepted: 27 February 2024     Published: 20 March 2024
Abstract

The representation of integers by prime factorization, proved by Euclid in the Fundamental Theorem of Arithmetic −also referred to as the Prime Factorization Theorem− although universal in scope, does not provide insight into the algebraic structure of primes themselves. No such insight is gained by summative prime factorization either, where a number can be represented as a sum of up to three primes, assuming Goldbach’s conjecture is true. In this paper, a third type of factorization is introduced, called hybrid prime factorization, defined as the representation of a number as sum −or difference− of two products of primes with no common factors between them. By using hybrid factorization, primes are expressed as algebraic functions of other primes, and primality is established by a single algebraic condition. Following a hybrid factorization approach, sufficient conditions for the existence of Goldbach pairs are derived, and their values are algebraically evaluated, based on the symmetry exhibited by Goldbach primes around their midpoint. Hybrid prime factorization is an effective way to represent, predict, compute, and analyze primes, expressed as algebraic functions. It is shown that the sequence of primes can be generated through an algebraic process with evolutionary properties. Since prime numbers do not follow any predetermined pattern, proving that they can be represented, computed and analyzed algebraically has important practical and theoretical ramifications.

 Published in Mathematics and Computer Science (Volume 9, Issue 1) DOI 10.11648/j.mcs.20240901.12 Page(s) 12-25 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2024. Published by Science Publishing Group
Keywords

Prime Number, Factorization, Optimization, Encoding, Goldbach Primes

References
 [1] Loconsole, M., Regolin, L. Are prime numbers special? Insights from the life sciences. Biol Direct 17, 11 (2022). https://doi.org/10.1186/s13062-022-00326-w [2] S. Torquato, G. Zhang and M. de Courcy-Ireland, “Uncovering multiscale order in the prime numbers via scattering”, Journal of Statistical Mechanics: Theory and Experiment, (2018) 093401, September 2018. https://doi.org/10.1088/1742-5468/aad6be [3] P. Billingsley, “Prime Numbers and Brownian Motion”, The American Mathematical Monthly, vol. 80, 1973, pp. 1099-1115. [Online]. Available: https://maa.org/programs/maa-awards/writing-awards/prime-numbers-and-brownian-motion [4] K. D. Thomas, “From Prime Numbers to Nuclear Physics and Beyond”, The Institute of Advanced Study (IAS), 2013. [Online]. Available: https://www.ias.edu/ideas/2013/primes-random-matrices [5] Wikipedia, "Montgomery's pair correlation conjecture" [Online]. Available: https://en.wikipedia.org/w/index.php?title=Montgomery%27s_pair_correlation_conjecture&oldid=1194579064 [6] H. Montgomery, “The pair of correlation of zeros of the zeta function”, Proc. Symp. Pure Math., 24: 181–193, 1973. [Online]. Available: https://websites.umich.edu/~hlm/paircor1.pdf [7] V. Barbarani, “A Quantum Model of the Distribution of Prime Numbers and the Riemann Hypothesis”, International Journal of Theoretical Physics (2020), 59: 2425–2470. https://doi.org/10.1007/s10773-020-04512-2 [8] J. I. Latorre and G. Sierra, “There is entanglement in the primes”, Quantum Information and Computation, 15. 10.26421/QIC15.7-8-6. https://doi.org/10.48550/arXiv.1403.4765 [9] A. Sugamoto, “Factorization of Number into Prime Numbers Viewed as Decay of Particle into Elementary Particles Conserving Energy”, Progress of Theoretical Physics 121 (2), February 2009, https://doi.org/10.1143/PTP.121.275 [10] M. Sanchis-Lozano, J. F. Barbero G. and J. Navarro-Salas, “Prime numbers, quantum field theory and the Goldbach conjecture”, International Journal of Modern Physics, Vol. 27 (23), September 2012. https://doi.org/10.1142/S0217751X12501369 [11] I. N. M. Papadakis, On the Universal Encoding Optimality of Primes. Mathematics 2021, 9 (24), 3155. https://doi.org/10.3390/math9243155 [12] C. K. Caldwell, The Prime Pages: Goldbach's conjecture. [Online]. Available: https://t5k.org/glossary/page.php?sort=GoldbachConjecture [13] Wikipedia, “Goldbach's conjecture” [Online]. Available: https://en.wikipedia.org/wiki/Goldbach%27s_conjecture [14] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, New York NY, USA: Oxford University Press, 2008 (6th edition), p. 23. [15] C. K. Caldwell, The Prime Pages: Prime Conjectures and Open Questions. [Online]. Available: https://t5k.org/notes/conjectures/ [16] T. Oliveira e Silva, S. Herzog and S. Pardi, “Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4∙1018”, Mathematics of Computation, 83 (2014), 2033-2060, November 2013. [Online]. Available: https://www.ams.org/journals/mcom/2014-83-288/S0025-5718-2013-02787-1/S0025-5718-2013-02787-1.pdf [17] Weisstein, Eric W. "Goldbach Partition." From MathWorld − A Wolfram Web Resource [Online]. Available: https://mathworld.wolfram.com/GoldbachPartition.html [18] R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Second Edition (2005), Springer, ISBN-13: 978-0387-25282-7. [19] Euclid, Elements, proposition 9.20 (p. 271) from the Greek text of J. L. Heiberg (1883-1885) based on Euclidis Elementa, edited and translated in English by Richard Fitzpatrick [Online]. Available: https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
• APA Style

Papadakis, I. N. M. (2024). Algebraic Representation of Primes by Hybrid Factorization. Mathematics and Computer Science, 9(1), 12-25. https://doi.org/10.11648/j.mcs.20240901.12

ACS Style

Papadakis, I. N. M. Algebraic Representation of Primes by Hybrid Factorization. Math. Comput. Sci. 2024, 9(1), 12-25. doi: 10.11648/j.mcs.20240901.12

AMA Style

Papadakis INM. Algebraic Representation of Primes by Hybrid Factorization. Math Comput Sci. 2024;9(1):12-25. doi: 10.11648/j.mcs.20240901.12

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volume = {9},
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abstract = {The representation of integers by prime factorization, proved by Euclid in the Fundamental Theorem of Arithmetic −also referred to as the Prime Factorization Theorem− although universal in scope, does not provide insight into the algebraic structure of primes themselves. No such insight is gained by summative prime factorization either, where a number can be represented as a sum of up to three primes, assuming Goldbach’s conjecture is true. In this paper, a third type of factorization is introduced, called hybrid prime factorization, defined as the representation of a number as sum −or difference− of two products of primes with no common factors between them. By using hybrid factorization, primes are expressed as algebraic functions of other primes, and primality is established by a single algebraic condition. Following a hybrid factorization approach, sufficient conditions for the existence of Goldbach pairs are derived, and their values are algebraically evaluated, based on the symmetry exhibited by Goldbach primes around their midpoint. Hybrid prime factorization is an effective way to represent, predict, compute, and analyze primes, expressed as algebraic functions. It is shown that the sequence of primes can be generated through an algebraic process with evolutionary properties. Since prime numbers do not follow any predetermined pattern, proving that they can be represented, computed and analyzed algebraically has important practical and theoretical ramifications.
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Author Information
• Independent Researcher, Charlotte, North Carolina, USA

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