On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation

Ioannis Papadakis

doi:doi:10.11648/j.mcs.20240905.12

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On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation

Ioannis Papadakis^*

Published in Mathematics and Computer Science (Volume 9, Issue 5)

Received: 27 October 2024 Accepted: 12 November 2024 Published: 29 November 2024

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Abstract

An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it is shown that prime factorization is an optimal projection operation on the set of integers; that Goldbach pairs correspond to solutions of an optimization problem; that hybrid prime factorization can be used to generate Goldbach primes; that prime symmetry, a powerful property of Goldbach primes, can be used to validate the binary Goldbach conjecture in short intervals, and to determine the rules that govern the “algebraic evolution” of Goldbach pairs, as the value of s increases; and that analytic approximation, using translational and rotational shifts of smooth functions, leads to a useful approximation of a primality test function and the prime counting function π(s). The paper’s findings support the broader hypothesis that prime numbers, by virtue of their optimality in representing, additively and multiplicatively, any measurable quantity in the universe, supported by the Fundamental Theorem of Arithmetic and the binary Goldbach conjecture, may be a viable alternative to the exclusive use of binary logic, as a means of achieving additional computational efficiencies of scale in the future.

Published in	Mathematics and Computer Science (Volume 9, Issue 5)
DOI	10.11648/j.mcs.20240905.12
Page(s)	96-113
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

Primes, Goldbach Conjecture, Projection, Optimization, Factorization, Prime Symmetry, Analytic Approximation

References

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[2]	E. W. Weisstein, Goldbach Conjecture. MathWorld-A Wolfram Web Resource. [Online]. Available https://mathworld.wolfram.com/GoldbachConjecture.html
[3]	C. K. Caldwell, The Prime Pages Goldbach's conjecture. [Online]. Available https://t5k.org/glossary/page.php?sort=GoldbachConjecture
[4]	C. K. Caldwell, The Prime Pages Prime Conjectures and Open Questions. [Online]. Available https://t5k.org/notes/conjectures/
[5]	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. Available https://www.ams.org/journals/mcom/2014-83-288/S0025-5718-2013-02787-1/S0025-5718-2013-02787-1.pdf
[6]	H. Montgomery and R. Vaughan, The exceptional set of Goldbach's problem. Acta Arithmetica 27.1 (1975) 353-370. http://eudml.org/doc/205349
[7]	H. A. Helfgott, The ternary Goldbach conjecture is true. (2013) arXiv 1312.7748. [Online]. Available https://arxiv.org/abs/1312.7748
[8]	H. A. Helfgott, The ternary Goldbach Problem, Ann. of Math. Studies (in publication). Available https://webusers.imj prg.fr/~harald.helfgott/anglais/publications.html
[9]	J. Derbyshire, Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics; Plume Washington, DC, USA; 2004.
[10]	H. M. Edwards, Riemann’s Zeta Function, New York NY, USA Academic Press, 1964 (Dover edition reprint, 2001), pp. 1-54.
[11]	B. Mazur and W. Stein, Prime Numbers and the Riemann Hypothesis, UK Cambridge University Press, 2017 (3^rd printing).
[12]	M. Cully-Hugill and A. W. Dudek, An explicit mean-value estimate for the Prime Number Theorem in intervals. Journal of the Australian Mathematical Society (2023), pp. 1–15. https://doi.org/10.1017/S1446788723000113
[13]	I. N. M. Papadakis, On the Universal Encoding Optimality of Primes. Mathematics 2021, 9 (24), 3155. https://doi.org/10.3390/math9243155
[14]	I. N. M. Papadakis, Algebraic Representation of Primes by Hybrid Factorization. Math. Comput. Sci. 2024, 9(1), 12-25. https://doi.org/10.11648/j.mcs.20240901.12
[15]	I. Papadakis, Representation and Generation of Prime and Coprime Numbers by Using Structured Algebraic Sums. Math. Comput. Sci. 2024, 9(3), 57-63. https://doi.org/10.11648/j.mcs.20240903.12
[16]	W. K. Nicholson, Linear Algebra with Applications, Lyryx Open Edition, p. 306 (Corollary 5.4.4). [Online]. Available https://lyryx.com/wp-content/uploads/2018/01/Nicholson-OpenLAWA-2018A.pdf
[17]	Wikipedia, Overdetermined system. [Online]. Available https://en.wikipedia.org/w/index.php?title=Overdetermined_system&oldid=1235933306
[18]	W. Rudin, Principles of Mathematical Analysis, McGraw-Hill New York, NY, USA; 1976 (3^rd edition), p. 228. [Online]. Available: https://www.gbv.de/dms/ilmenau/toc/254172466.PDF
[19]	C. K. Caldwell, The Prime Pages The first fifty million primes. [Online]. Available https://t5k.org/lists/small/millions/
[20]	H. Riesel and G. Göhl, Some Calculations Related to Riemann’s Prime Number Theorem. Mathematics of Computation 1970, vol. 24, no. 112, 969-983. https://doi.org/10.2307/2004630

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Papadakis, I. (2024). On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation. Mathematics and Computer Science, 9(5), 96-113. https://doi.org/10.11648/j.mcs.20240905.12

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ACS Style

Papadakis, I. On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation. Math. Comput. Sci. 2024, 9(5), 96-113. doi: 10.11648/j.mcs.20240905.12

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AMA Style

Papadakis I. On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation. Math Comput Sci. 2024;9(5):96-113. doi: 10.11648/j.mcs.20240905.12

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@article{10.11648/j.mcs.20240905.12,
  author = {Ioannis Papadakis},
  title = {On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation
},
  journal = {Mathematics and Computer Science},
  volume = {9},
  number = {5},
  pages = {96-113},
  doi = {10.11648/j.mcs.20240905.12},
  url = {https://doi.org/10.11648/j.mcs.20240905.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20240905.12},
  abstract = {An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it is shown that prime factorization is an optimal projection operation on the set of integers; that Goldbach pairs correspond to solutions of an optimization problem; that hybrid prime factorization can be used to generate Goldbach primes; that prime symmetry, a powerful property of Goldbach primes, can be used to validate the binary Goldbach conjecture in short intervals, and to determine the rules that govern the “algebraic evolution” of Goldbach pairs, as the value of s increases; and that analytic approximation, using translational and rotational shifts of smooth functions, leads to a useful approximation of a primality test function and the prime counting function π(s). The paper’s findings support the broader hypothesis that prime numbers, by virtue of their optimality in representing, additively and multiplicatively, any measurable quantity in the universe, supported by the Fundamental Theorem of Arithmetic and the binary Goldbach conjecture, may be a viable alternative to the exclusive use of binary logic, as a means of achieving additional computational efficiencies of scale in the future.
},
 year = {2024}
}

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T1  - On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation

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Y1  - 2024/11/29
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DO  - 10.11648/j.mcs.20240905.12
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VL  - 9
IS  - 5
ER  -

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Author Information

Ioannis Papadakis

Independent Researcher, Charlotte, United States of America

Contact Email

http://orcid.org/0000-0001-9245-0751

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APA Style

Papadakis, I. (2024). On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation. Mathematics and Computer Science, 9(5), 96-113. https://doi.org/10.11648/j.mcs.20240905.12

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ACS Style

Papadakis, I. On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation. Math. Comput. Sci. 2024, 9(5), 96-113. doi: 10.11648/j.mcs.20240905.12

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AMA Style

Papadakis I. On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation. Math Comput Sci. 2024;9(5):96-113. doi: 10.11648/j.mcs.20240905.12

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@article{10.11648/j.mcs.20240905.12,
  author = {Ioannis Papadakis},
  title = {On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation
},
  journal = {Mathematics and Computer Science},
  volume = {9},
  number = {5},
  pages = {96-113},
  doi = {10.11648/j.mcs.20240905.12},
  url = {https://doi.org/10.11648/j.mcs.20240905.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20240905.12},
  abstract = {An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it is shown that prime factorization is an optimal projection operation on the set of integers; that Goldbach pairs correspond to solutions of an optimization problem; that hybrid prime factorization can be used to generate Goldbach primes; that prime symmetry, a powerful property of Goldbach primes, can be used to validate the binary Goldbach conjecture in short intervals, and to determine the rules that govern the “algebraic evolution” of Goldbach pairs, as the value of s increases; and that analytic approximation, using translational and rotational shifts of smooth functions, leads to a useful approximation of a primality test function and the prime counting function π(s). The paper’s findings support the broader hypothesis that prime numbers, by virtue of their optimality in representing, additively and multiplicatively, any measurable quantity in the universe, supported by the Fundamental Theorem of Arithmetic and the binary Goldbach conjecture, may be a viable alternative to the exclusive use of binary logic, as a means of achieving additional computational efficiencies of scale in the future.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation

AU  - Ioannis Papadakis
Y1  - 2024/11/29
PY  - 2024
N1  - https://doi.org/10.11648/j.mcs.20240905.12
DO  - 10.11648/j.mcs.20240905.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 96
EP  - 113
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20240905.12
AB  - An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it is shown that prime factorization is an optimal projection operation on the set of integers; that Goldbach pairs correspond to solutions of an optimization problem; that hybrid prime factorization can be used to generate Goldbach primes; that prime symmetry, a powerful property of Goldbach primes, can be used to validate the binary Goldbach conjecture in short intervals, and to determine the rules that govern the “algebraic evolution” of Goldbach pairs, as the value of s increases; and that analytic approximation, using translational and rotational shifts of smooth functions, leads to a useful approximation of a primality test function and the prime counting function π(s). The paper’s findings support the broader hypothesis that prime numbers, by virtue of their optimality in representing, additively and multiplicatively, any measurable quantity in the universe, supported by the Fundamental Theorem of Arithmetic and the binary Goldbach conjecture, may be a viable alternative to the exclusive use of binary logic, as a means of achieving additional computational efficiencies of scale in the future.

VL  - 9
IS  - 5
ER  -

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