What question did this study set out to answer?

To model the conditional density of twin primes on the 6N±1 skeleton and validate the model through extensive computation.

June 2, 2026Open Access

Prime Distribution on the 6N Skeleton: A Factor-Resolved Local Model for the Conditional Density of Twin Primes, with Large-Scale Verification

Key Points

To model the conditional density of twin primes on the 6N±1 skeleton and validate the model through extensive computation.
Exhaustive computation of prime occurrence on the 6N±1 skeleton up to 10^10(1.67×10^9 centres scanned)
Identified 2.74×10^7 twin pairs and developed a factor-resolved local model for conditional density
Validation against ~10^8 nodes with accuracy observed/predicted = 1.000 ± 0.008
The model accurately reproduces the conditionally proportional twin prime density based on factorisation.
Achieved a worldwide validation across strata, showing a 3.3× enrichment range.
Demonstrated reproduction of squarefree bias reported previously to within -0.2% at the matching scale of 10^10.

Abstract

Exhaustive computation of prime occurrence on the 6N±1 skeleton up to 10¹0 (all 1. 67×10⁹ centres scanned; 2. 74×10⁷ twin pairs identified, shells S1–S10), together with a factor-resolved local model for the conditional density of twinprimes. Conditioning on the factorisation of the centre N, the conditional twindensity is shown to be proportional to E (N) = ∏ₐ|₍, ₐ>₃ (q−1) / (q−3). The modelis derived within the Hardy–Littlewood heuristic, validated factor by factoragainst ~10⁸ nodes (observed/predicted = 1. 000 ± 0. 008 across strata spanning a3. 3× enrichment range, selecting it over the naïve ∏ q/ (q−2) ), and shown toreproduce the squarefree bias reported by Puszkarz (2018, arXiv: 1807. 00406) towithin −0. 2% at the matching 10¹0 scale. This deposit contains: the five result tables (S1–S10, complete factorisation), the computation and figure-generation code, the two figures, and the manuscript (LaTeX + PDF). Scope. This is experimental/computational number theory. The model is avalidated Hardy–Littlewood heuristic, not an unconditional theorem; nothing herebears on the infinitude of twin primes. Priority for the underlying phenomenonbelongs to W. Puszkarz (2018) ; the cumulative twin counts coincide with OEISA007508.

Prime Distribution on the 6N Skeleton: A Factor-Resolved Local Model for the Conditional Density of Twin Primes, with Large-Scale Verification

Key Points

Abstract

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