Goldbach problems for Polignac families: twin, cousin and sexy summands
- 1Imperial College London
Abstract
For an even integer d, let PPd be the set of primes p such that p−d or p+d is also prime; thus PP2, PP4 and PP6 are the twin, cousin and sexy primes. We study the binary problem of representing an even integer n as p+q with p∈PPa and q∈PPb, which for a=b=2 is the twin-prime Goldbach problem of Zwillinger and Dubner. By an exact, integer-only witness search we determine the exceptional sets for all six pairings of twin, cousin and sexy primes up to 108, and for all $120$ pairs of gaps a,b≤ 30 up to 107. Every even n≥ 6 is the sum of a twin prime and a cousin prime in this range, whereas twin+twin and cousin+cousin have $34$ and $36$ exceptions, the largest being $4208$ and $2952$. We prove a structure theorem modulo~$3$ that explains this contrast: when 3nmid a, the representations of s, s+a, s+2a are locked together in the exact ratio 1:2:1, so equal-gap exceptions occur in arithmetic triples, while for a≠ b the class n≡ 0pmod 3 has two independent routes. As a corollary, the twin-prime Goldbach problem for multiples of $6$ is equivalent to a Goldbach problem for the index set {k:6k±1 both prime}. We formulate a Hardy--Littlewood type conjecture for ra,b(n), prove that its singular series is bounded below by an explicit absolute constant c0≈ 4.1512 (halved when a=b), and show that the prediction, refined by inclusion--exclusion, matches the data to within 1% near 107 and reproduces the number of observed exceptions. Any proof of the existence part for a single pair (a,b) would establish Polignac's conjecture for gap a or gap b.
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BibTeX
@misc{bilokon2026goldbach,
title = {{Goldbach problems for Polignac families: twin, cousin and sexy summands}},
author = {Paul Alexander Bilokon},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23463971},
url = {https://zenodo.org/records/23463971},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Goldbach problems for Polignac families: twin, cousin and sexy summands
AU - Paul Alexander Bilokon
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23463971
UR - https://zenodo.org/records/23463971
AB - For an even integer d, let PP_d be the set of primes p such that p−d or p+d is also prime; thus PP_2, PP_4 and PP_6 are the twin, cousin and sexy primes. We study the binary problem of representing an even integer n as p+q with p∈PP_a and q∈PP_b, which for a=b=2 is the twin-prime Goldbach problem of Zwillinger and Dubner. By an exact, integer-only witness search we determine the exceptional sets for all six pairings of twin, cousin and sexy primes up to 10^8, and for all $120$ pairs of gaps a,b≤ 30 up to 10^7. Every even n≥ 6 is the sum of a twin prime and a cousin prime in this range, whereas twin+twin and cousin+cousin have $34$ and $36$ exceptions, the largest being $4208$ and $2952$. We prove a structure theorem modulo~$3$ that explains this contrast: when 3nmid a, the representations of s, s+a, s+2a are locked together in the exact ratio 1:2:1, so equal-gap exceptions occur in arithmetic triples, while for a≠ b the class n≡ 0pmod 3 has two independent routes. As a corollary, the twin-prime Goldbach problem for multiples of $6$ is equivalent to a Goldbach problem for the index set {k:6k±1 both prime}. We formulate a Hardy--Littlewood type conjecture for r_a,b(n), prove that its singular series is bounded below by an explicit absolute constant c_0≈ 4.1512 (halved when a=b), and show that the prediction, refined by inclusion--exclusion, matches the data to within 1% near 10^7 and reproduces the number of observed exceptions. Any proof of the existence part for a single pair (a,b) would establish Polignac's conjecture for gap a or gap b.
ER -