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 = 2^{p-1}(2^p - 1)\\) iff \\(2^p - 1\\) is Mersenne prime. - New Mersenne prime: \\(p = 136279841\\), \\(M_p = 2^{136279841} - 1\\), 41,024,320 digits. - Odd perfect number: unknown existence; if exists, must be \\(>10^{1500}\\) (current lower bound) and have ≥101 prime factors with specific congruence constraints. - Claimed theorem: \\(P = 6^{m+1}N - 1\\) prime for \\(1 < N \\le 13\\), \\(N \\ne 8\\), \\(N \\ne i^{m+1} \\mod (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 = 2^{p-1}(2^p - 1)\\) are Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com