For N≥3, let ℒ(N) denote the primes in [3,2N-3]. The binary Goldbach condition at 2N is equivalent to positive prime incidence under reflection about the midpoint N. This paper studies prime incidence under two bounded integer-line isometries: (1) displaced reflection (reflected branch) and (2) translation (translated branch). In both cases, the prime and its image must remain in ℒ(N). For either branch, the Maximal Displacement Envelope (MDE) is the largest even displacement through which incidence remains positive at every even displacement, including zero; if incidence fails at zero, that branch has no MDE. When both branch envelopes exist, their minimum defines the combined MDE(N). The Prime Incidence Conjecture (PIC) asserts that MDE(N) exists for every N≥3 and that MDE(N)/N→2 as N→∞. At zero displacement, reflected incidence is precisely the binary Goldbach condition. Both branches admit single-parameter arithmetic formulations through the least prime realizing a prescribed even prime difference and the least symmetric prime-pair displacement about a midpoint. Exhaustive computation through N=3×107 determines the branch-specific and combined envelopes over the full admissible displacement range and shows that the normalized first-failure displacements trend toward the normalized endpoint 2-6/N over the computed range. Established analytic estimates further imply that, for every fixed 0<ε<2 and A>0, the number of even displacements s≤(2-ε)N at which at least one branch fails is OA,ε(N(log N)-A) for all sufficiently large N.
| Published in | Mathematics and Computer Science (Volume 11, Issue 6) |
| DOI | 10.11648/j.mcs.20261106.11 |
| Page(s) | 118-136 |
| 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 |
Binary Goldbach Conjecture, Prime Incidence, Prime Incidence Conjecture, Displacement, Reflection, Translation, Maximal Displacement Envelope, Left–Right Partition Table
| [1] | G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed. Oxford University Press, Oxford, 2008. |
| [2] | H. L. Montgomery and R. C. Vaughan, "The exceptional set of Goldbach's problem," Acta Arithmetica, vol. 27, pp. 353–370, 1975. |
| [3] | 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, vol. 83, no. 288, pp. 2033–2060, 2014. |
| [4] | G. Bhowmik and L. Grimmelt, "The exceptional set of the Goldbach problem," Analysis Mathematica, 2026. |
| [5] | Y. Tsuda, "Small gaps between primes that satisfy the Goldbach equation," Monatshefte für Mathematik, vol. 206, pp. 217–258, 2025. |
| [6] | I. Papadakis, "On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation," Mathematics and Computer Science, vol. 9, no. 5, pp. 96–113, 2024. |
| [7] | I. Papadakis, "Structural Failure Mode Analysis of the Binary Goldbach Conjecture," Mathematics and Computer Science, vol. 11, no. 2, pp. 17–32, 2026. |
| [8] | OEIS Foundation Inc., "A020483: Least prime p such that p+2n is also prime," The On-Line Encyclopedia of Integer Sequences. |
| [9] | OEIS Foundation Inc., "A047160: For n≥2, smallest m≥0 such that n-m and n+m are both primes, or -1 if no such m exists," The On-Line Encyclopedia of Integer Sequences |
| [10] | J. Pintz, "On the difference of primes," arXiv:1206.0149 [math.NT], 2012. |
| [11] | A. F. Lavrik, "On the theory of the distribution of sets of primes with given differences between them," Doklady Akademii Nauk SSSR, vol. 138, no. 6, pp. 1287–1290, 1961. |
| [12] | H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory II: Primes and Sieves, Cambridge Studies in Advanced Mathematics, vol. 218, Cambridge University Press, Cambridge, 2026. |
| [13] | G. Coppola and M. B. S. Laporta, "On the representation of even integers as sum of two almost equal primes," Rendiconti del Seminario Matematico dell'Universit`a e Politecnico di Torino, vol. 53, no. 3, pp. 245–252, 1995. |
| [14] | I. Papadakis, "On the Discrete and Continuous Harmonic Encoding of Primes," Mathematics and Computer Science, vol. 11, no. 3, pp. 45–56, 2026. |
| [15] | R. C. Baker, G. Harman, and J. Pintz, "The Difference Between Consecutive Primes, II," Proceedings of the London Mathematical Society, vol. 83, no. 3, pp. 532–562, 2001. |
| [16] | J. Nagura, "On the Interval Containing At Least One Prime Number," Proceedings of the Japan Academy, vol. 28, no. 4, pp. 177–181, 1952. |
APA Style
Papadakis, I. (2026). Prime Incidence Analysis Under Bounded Integer-Line Isometries. Mathematics and Computer Science, 11(6), 118-136. https://doi.org/10.11648/j.mcs.20261106.11
ACS Style
Papadakis, I. Prime Incidence Analysis Under Bounded Integer-Line Isometries. Math. Comput. Sci. 2026, 11(6), 118-136. doi: 10.11648/j.mcs.20261106.11
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Download@article{10.11648/j.mcs.20261106.11,
author = {Ioannis Papadakis},
title = {Prime Incidence Analysis Under Bounded Integer-Line Isometries},
journal = {Mathematics and Computer Science},
volume = {11},
number = {6},
pages = {118-136},
doi = {10.11648/j.mcs.20261106.11},
url = {https://doi.org/10.11648/j.mcs.20261106.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20261106.11},
abstract = {For N≥3, let ℒ(N) denote the primes in [3,2N-3]. The binary Goldbach condition at 2N is equivalent to positive prime incidence under reflection about the midpoint N. This paper studies prime incidence under two bounded integer-line isometries: (1) displaced reflection (reflected branch) and (2) translation (translated branch). In both cases, the prime and its image must remain in ℒ(N). For either branch, the Maximal Displacement Envelope (MDE) is the largest even displacement through which incidence remains positive at every even displacement, including zero; if incidence fails at zero, that branch has no MDE. When both branch envelopes exist, their minimum defines the combined MDE(N). The Prime Incidence Conjecture (PIC) asserts that MDE(N) exists for every N≥3 and that MDE(N)/N→2 as N→∞. At zero displacement, reflected incidence is precisely the binary Goldbach condition. Both branches admit single-parameter arithmetic formulations through the least prime realizing a prescribed even prime difference and the least symmetric prime-pair displacement about a midpoint. Exhaustive computation through N=3×107 determines the branch-specific and combined envelopes over the full admissible displacement range and shows that the normalized first-failure displacements trend toward the normalized endpoint 2-6/N over the computed range. Established analytic estimates further imply that, for every fixed 0<ε<2 and A>0, the number of even displacements s≤(2-ε)N at which at least one branch fails is OA,ε(N(log N)-A) for all sufficiently large N.},
year = {2026}
}
TY - JOUR T1 - Prime Incidence Analysis Under Bounded Integer-Line Isometries AU - Ioannis Papadakis Y1 - 2026/11/14 PY - 2026 N1 - https://doi.org/10.11648/j.mcs.20261106.11 DO - 10.11648/j.mcs.20261106.11 T2 - Mathematics and Computer Science JF - Mathematics and Computer Science JO - Mathematics and Computer Science SP - 118 EP - 136 PB - Science Publishing Group SN - 2575-6028 UR - https://doi.org/10.11648/j.mcs.20261106.11 AB - For N≥3, let ℒ(N) denote the primes in [3,2N-3]. The binary Goldbach condition at 2N is equivalent to positive prime incidence under reflection about the midpoint N. This paper studies prime incidence under two bounded integer-line isometries: (1) displaced reflection (reflected branch) and (2) translation (translated branch). In both cases, the prime and its image must remain in ℒ(N). For either branch, the Maximal Displacement Envelope (MDE) is the largest even displacement through which incidence remains positive at every even displacement, including zero; if incidence fails at zero, that branch has no MDE. When both branch envelopes exist, their minimum defines the combined MDE(N). The Prime Incidence Conjecture (PIC) asserts that MDE(N) exists for every N≥3 and that MDE(N)/N→2 as N→∞. At zero displacement, reflected incidence is precisely the binary Goldbach condition. Both branches admit single-parameter arithmetic formulations through the least prime realizing a prescribed even prime difference and the least symmetric prime-pair displacement about a midpoint. Exhaustive computation through N=3×107 determines the branch-specific and combined envelopes over the full admissible displacement range and shows that the normalized first-failure displacements trend toward the normalized endpoint 2-6/N over the computed range. Established analytic estimates further imply that, for every fixed 0<ε<2 and A>0, the number of even displacements s≤(2-ε)N at which at least one branch fails is OA,ε(N(log N)-A) for all sufficiently large N. VL - 11 IS - 6 ER -
Independent Researcher, Charlotte, United States of America
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