Twin Prime Progress: Bounded Gaps, Unproven Conjecture, and a Flawed Claim — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve methods and a claimed (likely flawed) constructive proof using Eratosthenes sieve bounds. | MATH: Twin primes: p, p+2 both prime. Maynard's theorem: liminfn→∞(pn+k−pn) < ∞ for any fixed k (bounded gaps for k-tuples). For k=1, best unconditional bound: gaps ≤ 246 infinitely often (Polymath8). Claimed arXiv proof (1708.07884) asserts at least 3 new twin pairs per increment of n when (6n+5)2 is used as a sieve bound — but this is not peer-validated and contradicts known density estimates. | CONNECTION: Twin primes modulo 6: all pairs > (3,5) are of form (6n−1, 6n+1) — a lattice of period 6, aligning with hexagonal (crystallographic) symmetry. The ratio of twin primes to primes ∼ 2C2/(log2 x) where C2 ≈ 0.66016 (twin prime constant). Note: C2 is not a golden-ratio multiple, but 2C2 ≈ 1.32032 — close to
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
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BibTeX
@misc{caldin2026twin,
title = {{Twin Prime Progress: Bounded Gaps, Unproven Conjecture, and a Flawed Claim — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23465072},
url = {https://zenodo.org/records/23465072},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twin Prime Progress: Bounded Gaps, Unproven Conjecture, and a Flawed Claim — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23465072
UR - https://zenodo.org/records/23465072
AB - FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve methods and a claimed (likely flawed) constructive proof using Eratosthenes sieve bounds. | MATH: Twin primes: p, p+2 both prime. Maynard's theorem: liminf_n→∞(p_n+k−p_n) < ∞ for any fixed k (bounded gaps for k-tuples). For k=1, best unconditional bound: gaps ≤ 246 infinitely often (Polymath8). Claimed arXiv proof (1708.07884) asserts at least 3 new twin pairs per increment of n when (6n+5)^2 is used as a sieve bound — but this is not peer-validated and contradicts known density estimates. | CONNECTION: Twin primes modulo 6: all pairs > (3,5) are of form (6n−1, 6n+1) — a lattice of period 6, aligning with hexagonal (crystallographic) symmetry. The ratio of twin primes to primes ∼ 2C_2/(log^2 x) where C_2 ≈ 0.66016 (twin prime constant). Note: C_2 is not a golden-ratio multiple, but 2C_2 ≈ 1.32032 — close to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -