{
  "abstract": "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 $p\\in\\PP_a$ and $q\\in\\PP_b$, 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 $a,b\\le 30$ up to $10^7$. Every even $n\\ge 6$ 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 $3\\nmid a$, 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 $a\\ne b$ the class $n\\equiv 0\\pmod 3$ 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 $\\{k:6k\\pm1\\text{ both prime}\\}$. 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 $c_0\\approx 4.1512$ (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 $b$.",
  "abstract_shown": true,
  "affiliations": [
    "Imperial College London"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Paul Alexander Bilokon",
      "orcid": "0000-0001-6846-6649"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5q8b9xxyea2n.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23463971",
  "fulltext_url": null,
  "id": "e5q8b9xxyea2n",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23463971",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23463971/files/polignac_goldbach.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5q8b9xxyea2n.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:20:46Z",
      "native_id": "23463971",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23463971"
    }
  ],
  "subjects": [],
  "title": "Goldbach problems for Polignac families: twin, cousin and sexy summands",
  "title_text": "Goldbach problems for Polignac families: twin, cousin and sexy summands",
  "type": "preprint",
  "updated_at": "2026-11-02T12:20:46Z",
  "url": "https://research.draconitic.com/paper/e5q8b9xxyea2n",
  "venue": "Zenodo",
  "versions": []
}