Perfect Numbers, Mersenne Primes, and the Unresolved Odd Case — E8 Intelligence Research

Abstract

FINDING: Perfect numbers remain tied to even Mersenne primes (Euclid–Euler); the largest known prime is 2^136279841 − 1 (41,024,320 digits); odd perfect numbers remain unproven; a claimed 6^m·N − 1 prime theorem is unverified and likely flawed. | MATH: Even perfect number: P = 2^(p−1)(2^p − 1) where (2^p − 1) is Mersenne prime. New prime: p = 136,279,841 → digits = floor(p·log10(2)) + 1 = 41,024,320. Odd perfect number: if exists, must be > 10^1500 (known lower bound), form N = q^α · ∏ p_i^(2β_i) with q ≡ α ≡ 1 (mod 4). The arXiv claim: P = 6^(m+1)·N − 1 prime for 1 < N ≤ 13, N ≠ 8, N ≠ i^(m+1) mod (6i+1) — this is ad hoc, not a theorem; no rigorous proof of infinitude. | CONNECTION: Mersenne primes correspond to even perfect numbers; the exponent p must be prime. The structure of even perfect numbers is triangular: P = 1 + 2 + ... + (2^p − 1) — a triangular number. The ratio of consecutive perfect numbers grows as ~2^p, not a fixed golden ratio. However, the sum of reciprocals of all Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com