New Mersenne Prime Record and the Unresolved Odd Perfect Number Question — E8 Intelligence Research

FINDING: Perfect numbers remain tied to Mersenne primes (Euclid–Euler), with a new record prime 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: - Euclid–Euler: Even perfect number \\(P = 2^{p-1}(2^p - 1)\\) iff \\(2^p - 1\\) is prime (Mersenne prime). - New Mersenne prime: \\(p = 136279841\\), so \\(M_p = 2^{136279841} - 1\\). - Odd perfect number: if exists, \\(N = q^\\alpha \\prod p_i^{2\\beta_i}\\), with \\(q \\equiv \\alpha \\equiv 1 \\pmod{4}\\) (Euler form). - Claimed theorem: \\(P = 6^{m+1}N - 1\\) prime for \\(1 < N \\le 13\\), \\(N \\ne 8\\), \\(N \\ne i^{m+1} \\bmod (6i+1)\\) — this is ad hoc, not a general result; no known proof of infinite Mersenne primes. CONNECTION: - Mersenne primes \\(M_p\\) relate to binary repunits: \\(M_p = 111...111_2\\) (p ones) — a lattice of dimension p in binary space. - Perfect numbers \\(P = 2^{p-1}M_p\\) have divisors summing to \\(2P\\) — a symmetry under the divisor l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com