{
  "abstract": "We give a short autonomous recurrence that generates every odd prime, in increasing order, from the single initial value $8$. Its integer state determines both its output and its next update: the lowest set bit gives the next prime, while the highest set bit determines the sieve period. The same state also determines the following prime gap. A related recurrence represents the prime-counting function as the $2$-adic valuation of a single integer. Independently, a direct packing of a finite divisor sieve yields fixed arithmetic constructions for the $n$-th prime, the prime-counting function, and the twin-prime counting function. We express these constructions as straight-line programs using addition, truncated subtraction, multiplication, integer division, exponentiation, and remainder. Counting operations with reuse of computed values gives programs of lengths $166$, $132$, and $167$, respectively; one autonomous recurrence update uses $99$ operations. Explicit arithmetic subroutines remove all auxiliary operations, including bitwise AND. The programs unfold into fixed arithmetic terms without variable-length sums, products, or searches.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13365",
  "author_total": 2,
  "authors": [
    {
      "name": "Joseph M. Shunia"
    },
    {
      "name": "Mihai Prunescu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7as00r64asch4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13365",
  "fulltext_url": null,
  "id": "7as00r64asch4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13365",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221511139",
  "pdf_url": "https://arxiv.org/pdf/2610.13365",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7as00r64asch4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13365",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13365"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Short Recurrences and Straight-Line Programs for Primes",
  "title_text": "Short Recurrences and Straight-Line Programs for Primes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/7as00r64asch4",
  "venue": "arXiv",
  "versions": []
}