A Deterministic Prime-Generating Chain from Rowland-Type Recurrences
Year: 2026
Type: preprint
Abstract: We study a deterministic prime-generating chain derived from Eric Rowland'sprime-generating recurrence a(n) = a(n−1) + gcd(n, a(n−1)) (2008). Once therecurrence locks into its safe invariant regime, its record emissions obey atime-free recurrence Φ that maps one prime to the next: from a single seedprime p₀, the chain produces an unbounded sequence of primes with no sieve,no candidate search, and no randomness. We give compact proofs of thestructural lemmas involved (distance monotonicity, escape coprimality,parity invariance, and an exact fire-value identity that closes the s² edgecase), show that Φ reproduces the record sequence of the full orbit exactly,and establish a doubling growth law with shrinking correction terms.We then map the functional graph of Φ over thousands of seeds: it is acycle-free forest of in-trees whose mergers are heavily front-loaded — ashell-Bernoulli model built exclusively from public ingredients (π(x) and ameasured growth constant) quantifies the effect (κ_early ≈ 1.25 vs.κ_late < 0.25 at 95% confidence), and dynasty count grows near-linearly withthe seed range. Finally we quantify honest limits: per-step termination is anopen Cramér-type gap problem, factoring costs inside the hop rule cappractical reachability near 30 decimal digits, chains cover only anexponentially thin subset of all primes, and the machinery contributesexactly nothing to RSA factoring. A methodological post-mortem documents howan apparent 2–3× twin-prime excess dissolved against exact empiricalbaselines, leaving a small genuine residue tied to residue classes of thefinal hop.
Author: Stankusch, L
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204135762
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