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

FINDING: Perfect numbers remain tied to even Mersenne primes (Euclid-Euler); odd perfect numbers still unproven; new largest Mersenne prime 2^136279841 − 1 (41,024,320 digits) discovered. | MATH: Even perfect number N = 2^(p−1)(2^p − 1) where (2^p − 1) is prime (Mersenne). New prime: p = 136,279,841; N has 41,024,320 digits. Odd perfect number existence: open (if exists, must be > 10^1500, have ≥ 101 prime factors, largest prime factor > 10^8). | CONNECTION: Mersenne primes correspond to even perfect numbers via binary repunit structure: 2^p − 1 = (111...1)_2 with p ones. Perfect numbers in binary: N = (111...1)(000...0) — a palindromic block structure. This mirrors crystallographic root system A_p (dimension p) where the sum of positive roots = p(p+1)/2 — a triangular number, and perfect numbers are a subset of triangular numbers (T2^p−1 = N). No direct golden ratio or base-60 link; but the binary symmetry (all ones then all zeros) echoes lattice periodicity. | DEPTH: 6 — Profound Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com