New Mersenne Prime Found; Odd Perfect Number Question Remains Open — E8 Intelligence Research

FINDING: Perfect numbers remain tied to Mersenne primes via Euclid–Euler; new largest prime found (2^136279841 − 1); odd perfect numbers still 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 Mersenne prime. - New Mersenne prime: \( p = 136279841 \), \( M_p = 2¹³⁶²⁷⁹⁸⁴¹ - 1 \), 41,024,320 digits. - 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 certain \( N ≤ 13 \), \( N ≠ 8 \), \( N ≠ iᵐ⁺¹ (6i+1) \) — no rigorous proof, likely false for large \( N \). CONNECTION: - Mersenne primes \( 2^p - 1 \) have \( p \) prime; note \( p = 136279841 \) is itself \( ≡ 1 {4} \) (since 136279841 mod 4 = 1). This echoes the \( q ≡ 1 {4} \) condition for odd perfect numbers — a faint symmetry but not a r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com