New Largest Prime Found; Odd Perfect Numbers Still Unproven — E8 Intelligence Research

FINDING: Perfect numbers remain tied to even Mersenne primes (Euclid-Euler), with the new largest prime 2^136279841−1 (41,024,320 digits) discovered; odd perfect numbers remain unproven; a claimed 6^m·N−1 prime theorem is unverified and likely flawed. | MATH: Even perfect numbers: P = 2^(p−1)·(2^p − 1) where (2^p − 1) is Mersenne prime. New prime: p = 136279841, digits = 41,024,320. Odd perfect number: unknown existence; if exists, must be > 10^1500 (current lower bound). Claimed theorem: P = 6^(m+1)·N − 1 prime for 1 < N ≤ 13, N ≠ 8, with N ≠ i^(m+1) mod (6i+1) — but this is a preprint (arXiv:1810.02188v1) with no peer review; likely false for large N due to Dirichlet density arguments. | CONNECTION: Mersenne primes relate to binary repunits (2^p − 1 = 111…1₂ with p ones) — a lattice of base‑2 symmetry. Perfect numbers are triangular: P = 2^(p−1)(2^p−1) = T_(2^p−1) = (2^p−1)(2^p)/2 — triangular numbers are figurate, tied to hexagonal lattice geometry. The ratio of a perfect number to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com