Perfect Numbers, Mersenne Primes, and a Dubious Primality Claim — E8 Intelligence Research

FINDING: Perfect numbers remain tied to even Mersenne primes (Euclid-Euler), odd perfect numbers remain unproven, and a new largest prime (2^136279841 − 1) was discovered; a claimed 6^m·N − 1 primality theorem is dubious. MATH: - Euclid-Euler: Even perfect number \(P = 2ᵖ⁻¹(2^p - 1)\) iff \(2^p - 1\) is Mersenne prime. - New Mersenne prime: \(p = 136279841\), \(M_p = 2¹³⁶²⁷⁹⁸⁴¹ - 1\), 41,024,320 digits. - Odd perfect number: unknown existence; if exists, must be \(>10¹⁵⁰⁰\) (current lower bound) and have ≥101 prime factors with specific congruence constraints. - Claimed theorem: \(P = 6ᵐ⁺¹N - 1\) prime for \(1 < N ≤ 13\), \(N ≠ 8\), \(N ≠ iᵐ⁺¹ (6i+1)\), \(m\) odd positive — this is **not** a proven general result; the arXiv paper (1810.02188v1) has known flaws (counterexamples exist for larger N). CONNECTION: - Mersenne primes \(M_p = 2^p - 1\) relate to binary repunits — base-2, not base-60. However, perfect numbers \(P = 2ᵖ⁻¹(2^p - 1)\) are Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com