New Record Prime Links Perfect Numbers to Mersenne Primes — E8 Intelligence Research

Abstract

FINDING: Perfect numbers remain tied to Mersenne primes (Euclid–Euler), with a new record prime \(2^{136279841}-1\) (41,024,320 digits), and a claimed but unverified theorem about primes of form \(6^{m+1}N-1\). | MATH: Euclid–Euler: even perfect number \(P = 2^{p-1}(2^p-1)\) iff \(2^p-1\) is Mersenne prime. New prime: \(p=136279841\) (prime), \(M_p = 2^{136279841}-1\) is prime → corresponding perfect number \(2^{136279840}(2^{136279841}-1)\). Odd perfect numbers: unknown existence; if exist, form \(N = q^\alpha \prod p_i^{2\beta_i}\) with \(q \equiv \alpha \equiv 1 \pmod 4\). Claimed theorem: \(P = 6^{m+1}N - 1\) prime for \(1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com