New Mersenne Prime Discovery and Unverified 6m-Form Theorem Link — E8 Intelligence Research

FINDING: Perfect numbers remain tied to Mersenne primes via the Euclid–Euler theorem; a new Mersenne prime (2^136279841 − 1) was discovered, and a claimed 6m-form prime theorem exists but is unverified. MATH: - Euclid–Euler: If \(2^p − 1\) is prime (Mersenne prime), then \(N = 2ᵖ⁻¹(2^p − 1)\) is an even perfect number. Converse (Euler): every even perfect number has this form. - New Mersenne prime: \(p = 136279841\), \(M_p = 2¹³⁶²⁷⁹⁸⁴¹ − 1\), 41,024,320 digits. - Odd perfect numbers: none known; existence is an open problem (oldest unsolved in number theory). - Claimed theorem (arXiv 1810.02188): \(P = 6ᵐ⁺¹ · N − 1\) prime for certain \(N ≤ 13\), \(N ≠ 8\), with exclusion \(N ≠ iᵐ⁺¹ (6i+1)\). This is not peer‑reviewed and likely flawed; no known structural breakthrough. CONNECTION: - Perfect numbers are highly composite; their divisors sum to the number — a multiplicative harmony. - The form \(2ᵖ⁻¹(2^p − 1)\) involves powers of 2, not dire Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com