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 \( P = 2^{p-1}(2^p - 1) \) iff \( 2^p - 1 \) is Mersenne prime. - New Mersenne prime: \( p = 136279841 \), \( M_p = 2^{136279841} - 1 \), 41,024,320 digits. - Odd perfect number: 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 certain \( N \le 13 \), \( N \ne 8 \), \( N \ne i^{m+1} \bmod (6i+1) \) — no rigorous proof, likely false for large \( N \). CONNECTION: - Mersenne primes \( 2^p - 1 \) have \( p \) prime; note \( p = 136279841 \) is itself \( \equiv 1 \pmod{4} \) (since 136279841 mod 4 = 1). This echoes the \( q \equiv 1 \pmod{4} \) 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