On the Discrete and Continuous Harmonic Encoding of Primes

Ioannis Papadakis

doi:doi:10.11648/j.mcs.20261103.11

Research Article |

| Peer-Reviewed

On the Discrete and Continuous Harmonic Encoding of Primes

Ioannis Papadakis^*

Published in Mathematics and Computer Science (Volume 11, Issue 3)

Received: 4 April 2026 Accepted: 15 April 2026 Published: 21 May 2026

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The dichotomy between the prime-based multiplicative and additive representations of integers poses fundamental and distinct challenges in analyzing the underlying prime distribution. This paper contributes to this area of research by introducing a tripartite framework for the lossless mathematical encoding of primes and their additive partitions. First, we establish Hybrid Prime Factorization (HPF) as a bounded a priori structural prime-generation framework. On certified intervals, primality is forced by disjoint partitions of canonical prime bases under the stated magnitude bound, so that certain HPF configurations yield prime outputs structurally, i.e., without requiring a separate post hoc primality test of the evaluated output. Second, to address the linear expansion of additive partitions, we introduce a deterministic pairing map, L(N), which losslessly compresses the entire additive state of even integer partitions into a single, uniquely factorizable scalar in ℤ. Finally, recognizing the asymptotically factorial limits of discrete integer representation, we map this arithmetic complexity into the continuous domain. We derive bounded, piecewise smooth harmonic sieve functions over ℝ \ ℤ that isolate prime and composite structures through the limits of indeterminate trigonometric forms. This progression establishes that prime complexity need not be confined to discrete combinatorial bounds, but can be translated into continuous harmonic functions, demonstrating that the prime counting function π(x) can be generated as a sum of continuous harmonic trigonometric functions.

Published in	Mathematics and Computer Science (Volume 11, Issue 3)
DOI	10.11648/j.mcs.20261103.11
Page(s)	45-56
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), 2026. Published by Science Publishing Group

Keywords

Prime Distribution, Hybrid Prime Factorization, Lossless Encoding, Continuous Harmonic Sieves, Analytic Number Theory, Additive Prime Partitions, Goldbach Pairings

References

[1]	Ribenboim, P. The New Book of Prime Number Records, 3rd ed.; Springer-Verlag: New York, NY, USA, 1996. https://doi.org/10.1007/978-1-4612-0759-7
[2]	Helfgott, H. A. The ternary Goldbach conjecture is true. arXiv 2013, arXiv: 1312.7748. https://doi.org/10.48550/arXiv.1312.7748
[3]	Dudley, U. Formulas for Primes. Mathematics Magazine 1983, 56(1), 17–22. https://doi.org/10.1080/0025570X.1983.11977009
[4]	Dudley, U. History of a Formula for Primes. The American Mathematical Monthly 1969, 76(1), 23–28. https://doi.org/10.1080/00029890.1969.12000121
[5]	Papadakis, I. On the Universal Encoding Optimality of Primes. Mathematics 2021, 9, 3155. https://doi.org/10.3390/math9243155
[6]	Papadakis, I. Structural Failure Mode Analysis of the Binary Goldbach Conjecture. Mathematics and Computer Science 2026, 11(2), 17–32. https://doi.org/10.11648/j.mcs.20261102.11
[7]	Papadakis, I. Representation and Generation of Prime and Coprime Numbers by Using Structured Algebraic Sums. Mathematics and Computer Science 2024, 9(3), 57–63. https://doi.org/10.11648/j.mcs.20240903.12
[8]	Papadakis, I. Algebraic Representation of Primes by Hybrid Factorization. Mathematics and Computer Science 2024, 9(1), 12–25. https://doi.org/10.11648/j.mcs.20240901.12
[9]	Mahler, K. Zur Approximation algebraischer Zahlen. I.(Über den größten Primteiler binärer Formen). Mathematische Annalen 1933, 107, 691–730.
[10]	Shorey, T. N.; Tijdeman, R. Exponential Diophantine Equations; Cambridge Tracts in Mathematics, Vol. 87; Cambridge University Press: Cambridge, UK, 1986. https://doi.org/10.1017/CBO9780511566042
[11]	Papadakis, I. On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation. Mathematics and Computer Science 2024, 9(5), 96–113. https://doi.org/10.11648/j.mcs.20240905.12
[12]	Lewis, H. R.; Papadimitriou, C. H. Elements of the Theory of Computation, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 1997.
[13]	Gödel, K. On formally undecidable propositions of Principia Mathematica and related systems I. In From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931; van Heijenoort, J., Ed.; Harvard University Press: Cambridge, MA, USA, 1967; pp. 596–616. https://doi.org/10.4159/harvard.9780674866620.c44
[14]	Montgomery, H. L. Topics in Multiplicative Number Theory, Vol. 227. Springer-Verlag: Berlin/Heidelberg, Germany, 1971. https://doi.org/10.1007/BFb0060851
[15]	Tao, T. Higher Order Fourier Analysis, Vol. 142. American Mathematical Society: Providence, RI, USA, 2012. https://doi.org/10.1090/gsm/142
[16]	Kolmogorov, A. N. Three approaches to the quantitative definition of information. Problems of Information Transmission 1965, 1(1), 1–7.

Cite This Article

Plain Text BibTeX RIS

APA Style

Papadakis, I. (2026). On the Discrete and Continuous Harmonic Encoding of Primes. Mathematics and Computer Science, 11(3), 45-56. https://doi.org/10.11648/j.mcs.20261103.11

Copy | Download

ACS Style

Papadakis, I. On the Discrete and Continuous Harmonic Encoding of Primes. Math. Comput. Sci. 2026, 11(3), 45-56. doi: 10.11648/j.mcs.20261103.11

Copy | Download

AMA Style

Papadakis I. On the Discrete and Continuous Harmonic Encoding of Primes. Math Comput Sci. 2026;11(3):45-56. doi: 10.11648/j.mcs.20261103.11

Copy | Download

@article{10.11648/j.mcs.20261103.11,
  author = {Ioannis Papadakis},
  title = {On the Discrete and Continuous Harmonic Encoding of Primes
},
  journal = {Mathematics and Computer Science},
  volume = {11},
  number = {3},
  pages = {45-56},
  doi = {10.11648/j.mcs.20261103.11},
  url = {https://doi.org/10.11648/j.mcs.20261103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20261103.11},
  abstract = {The dichotomy between the prime-based multiplicative and additive representations of integers poses fundamental and distinct challenges in analyzing the underlying prime distribution. This paper contributes to this area of research by introducing a tripartite framework for the lossless mathematical encoding of primes and their additive partitions. First, we establish Hybrid Prime Factorization (HPF) as a bounded a priori structural prime-generation framework. On certified intervals, primality is forced by disjoint partitions of canonical prime bases under the stated magnitude bound, so that certain HPF configurations yield prime outputs structurally, i.e., without requiring a separate post hoc primality test of the evaluated output. Second, to address the linear expansion of additive partitions, we introduce a deterministic pairing map, L(N), which losslessly
compresses the entire additive state of even integer partitions into a single, uniquely factorizable scalar in ℤ. Finally, recognizing the asymptotically factorial limits of discrete integer representation, we map this arithmetic complexity into the
continuous domain. We derive bounded, piecewise smooth harmonic sieve functions over ℝ \ ℤ that isolate prime and composite structures through the limits of indeterminate trigonometric forms. This progression establishes that prime complexity need not be confined to discrete combinatorial bounds, but can be translated into continuous harmonic functions, demonstrating that the prime counting function π(x) can be generated as a sum of continuous harmonic trigonometric functions.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - On the Discrete and Continuous Harmonic Encoding of Primes

AU  - Ioannis Papadakis
Y1  - 2026/05/21
PY  - 2026
N1  - https://doi.org/10.11648/j.mcs.20261103.11
DO  - 10.11648/j.mcs.20261103.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 45
EP  - 56
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20261103.11
AB  - The dichotomy between the prime-based multiplicative and additive representations of integers poses fundamental and distinct challenges in analyzing the underlying prime distribution. This paper contributes to this area of research by introducing a tripartite framework for the lossless mathematical encoding of primes and their additive partitions. First, we establish Hybrid Prime Factorization (HPF) as a bounded a priori structural prime-generation framework. On certified intervals, primality is forced by disjoint partitions of canonical prime bases under the stated magnitude bound, so that certain HPF configurations yield prime outputs structurally, i.e., without requiring a separate post hoc primality test of the evaluated output. Second, to address the linear expansion of additive partitions, we introduce a deterministic pairing map, L(N), which losslessly
compresses the entire additive state of even integer partitions into a single, uniquely factorizable scalar in ℤ. Finally, recognizing the asymptotically factorial limits of discrete integer representation, we map this arithmetic complexity into the
continuous domain. We derive bounded, piecewise smooth harmonic sieve functions over ℝ \ ℤ that isolate prime and composite structures through the limits of indeterminate trigonometric forms. This progression establishes that prime complexity need not be confined to discrete combinatorial bounds, but can be translated into continuous harmonic functions, demonstrating that the prime counting function π(x) can be generated as a sum of continuous harmonic trigonometric functions.
VL  - 11
IS  - 3
ER  -

Copy | Download

Author Information

Ioannis Papadakis

Independent Researcher, Charlotte, USA

Contact Email

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

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Papadakis, I. (2026). On the Discrete and Continuous Harmonic Encoding of Primes. Mathematics and Computer Science, 11(3), 45-56. https://doi.org/10.11648/j.mcs.20261103.11

Copy | Download

ACS Style

Papadakis, I. On the Discrete and Continuous Harmonic Encoding of Primes. Math. Comput. Sci. 2026, 11(3), 45-56. doi: 10.11648/j.mcs.20261103.11

Copy | Download

AMA Style

Papadakis I. On the Discrete and Continuous Harmonic Encoding of Primes. Math Comput Sci. 2026;11(3):45-56. doi: 10.11648/j.mcs.20261103.11

Copy | Download

@article{10.11648/j.mcs.20261103.11,
  author = {Ioannis Papadakis},
  title = {On the Discrete and Continuous Harmonic Encoding of Primes
},
  journal = {Mathematics and Computer Science},
  volume = {11},
  number = {3},
  pages = {45-56},
  doi = {10.11648/j.mcs.20261103.11},
  url = {https://doi.org/10.11648/j.mcs.20261103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20261103.11},
  abstract = {The dichotomy between the prime-based multiplicative and additive representations of integers poses fundamental and distinct challenges in analyzing the underlying prime distribution. This paper contributes to this area of research by introducing a tripartite framework for the lossless mathematical encoding of primes and their additive partitions. First, we establish Hybrid Prime Factorization (HPF) as a bounded a priori structural prime-generation framework. On certified intervals, primality is forced by disjoint partitions of canonical prime bases under the stated magnitude bound, so that certain HPF configurations yield prime outputs structurally, i.e., without requiring a separate post hoc primality test of the evaluated output. Second, to address the linear expansion of additive partitions, we introduce a deterministic pairing map, L(N), which losslessly
compresses the entire additive state of even integer partitions into a single, uniquely factorizable scalar in ℤ. Finally, recognizing the asymptotically factorial limits of discrete integer representation, we map this arithmetic complexity into the
continuous domain. We derive bounded, piecewise smooth harmonic sieve functions over ℝ \ ℤ that isolate prime and composite structures through the limits of indeterminate trigonometric forms. This progression establishes that prime complexity need not be confined to discrete combinatorial bounds, but can be translated into continuous harmonic functions, demonstrating that the prime counting function π(x) can be generated as a sum of continuous harmonic trigonometric functions.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - On the Discrete and Continuous Harmonic Encoding of Primes

AU  - Ioannis Papadakis
Y1  - 2026/05/21
PY  - 2026
N1  - https://doi.org/10.11648/j.mcs.20261103.11
DO  - 10.11648/j.mcs.20261103.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 45
EP  - 56
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20261103.11
AB  - The dichotomy between the prime-based multiplicative and additive representations of integers poses fundamental and distinct challenges in analyzing the underlying prime distribution. This paper contributes to this area of research by introducing a tripartite framework for the lossless mathematical encoding of primes and their additive partitions. First, we establish Hybrid Prime Factorization (HPF) as a bounded a priori structural prime-generation framework. On certified intervals, primality is forced by disjoint partitions of canonical prime bases under the stated magnitude bound, so that certain HPF configurations yield prime outputs structurally, i.e., without requiring a separate post hoc primality test of the evaluated output. Second, to address the linear expansion of additive partitions, we introduce a deterministic pairing map, L(N), which losslessly
compresses the entire additive state of even integer partitions into a single, uniquely factorizable scalar in ℤ. Finally, recognizing the asymptotically factorial limits of discrete integer representation, we map this arithmetic complexity into the
continuous domain. We derive bounded, piecewise smooth harmonic sieve functions over ℝ \ ℤ that isolate prime and composite structures through the limits of indeterminate trigonometric forms. This progression establishes that prime complexity need not be confined to discrete combinatorial bounds, but can be translated into continuous harmonic functions, demonstrating that the prime counting function π(x) can be generated as a sum of continuous harmonic trigonometric functions.
VL  - 11
IS  - 3
ER  -

Copy | Download