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 \(P = 2ᵖ⁻¹(2^p - 1)\) iff \(2^p - 1\) is prime (Mersenne prime). - New Mersenne prime: \(p = 136279841\), so \(M_p = 2¹³⁶²⁷⁹⁸⁴¹ - 1\). - Odd perfect number: if exists, \(N = q^α ∏ p_i2β_i\), with \(q ≡ α ≡ 1 {4}\) (Euler form). - Claimed theorem: \(P = 6ᵐ⁺¹N - 1\) prime for \(1 < N ≤ 13\), \(N ≠ 8\), \(N ≠ iᵐ⁺¹ (6i+1)\) — 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 \(P = 2ᵖ⁻¹M_p\) 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