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

FINDING: Perfect numbers remain tied to Mersenne primes via Euclid–Euler; new largest Mersenne prime 2^136279841 − 1 (41,024,320 digits) confirmed; odd perfect numbers still unproven; a claimed 6^m·N − 1 prime theorem is mathematically suspect. | MATH: Even perfect numbers: P = 2^(p−1)·(2^p − 1) where (2^p − 1) is Mersenne prime. New prime: p = 136,279,841 → P has 41,024,320 digits. Euclid–Euler theorem: all even perfects are of this form. Odd perfects: if N odd perfect, N = q^α · ∏ p_i^(2e_i) with q ≡ 1 (mod 4), α ≡ 1 (mod 4) — necessary conditions only. The arXiv claim (P = 6^(m+1)·N − 1) is not a theorem; it fails for N=8 excluded, but the modular condition is ad hoc and unproven for N>13. | CONNECTION: Perfect numbers are triangular: P = 2^(p−1)(2^p − 1) = T_(2^p − 1) = (2^p − 1)(2^p)/2 — a triangular number, linking to hexagonal numbers (2^p − 1 is hexagonal). The ratio of P to its largest proper divisor (2^p − 1) is 2^(p−1), a power of 2 — binary symmetry. No direct golden ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com