Euclid-Euler Theorem: Binary Scaling Links Even Perfect Numbers to Mersenne Primes — E8 Intelligence Research

FINDING: Euclid-Euler theorem establishes a bijection between even perfect numbers and Mersenne primes, revealing a deep binary-scaling structure in arithmetic perfection. | MATH: Even perfect number \(N = 2ᵖ⁻¹(2^p - 1)\) where \(2^p - 1\) is prime (Mersenne prime). Equivalently, \(σ(N) = 2N\) (sum of divisors). The theorem: \(N\) is even perfect iff \(N = 2ᵖ⁻¹M_p\) with \(M_p = 2^p - 1\) prime. | CONNECTION: The factor \(2ᵖ⁻¹\) is a pure binary shift — a scaling symmetry in base-2. The Mersenne prime \(M_p = 2^p - 1\) is the largest power of 2 minus 1, linking to binary repunits. The ratio \(N / 2ᵖ⁻¹ = M_p\) — the "defect" from the next power of 2. In geometric terms, \(2ᵖ⁻¹\) is the order of the hyperoctahedral group's binary subgroup; the structure echoes root lattice \(Aₚ₋₁\) or \(D_p\) scaling. The perfect number itself, e.g., 6 = 2·3, 28 = 4·7, 496 = 16·31, 8128 = 64·127 — each is \(2ᵏ(2ᵏ⁺¹-1)\) — a product of a power of 2 and a Mersenne prime. The ra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com