Euclid-Euler Theorem: Binary-Scaling Symmetry 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 symmetry in the divisor function. | MATH: Even perfect number \(N = 2^{p-1}(2^p - 1)\) where \(2^p - 1\) is prime (Mersenne prime). Equivalently, \(\sigma(N) = 2N\) (σ = sum of divisors). The binary representation of \(N\) is \(p\) ones followed by \(p-1\) zeros — a self-similar block structure. | CONNECTION: The form \(2^{p-1}(2^p - 1)\) exhibits a **geometric ratio** of \(2^p - 1\) to \(2^{p-1}\), which for large \(p\) approaches \(2\) — but more strikingly, the **fractional part** of \(N/(2^p - 1)^2\) is \(2^{-(p+1)}\), linking to dyadic scaling. The **sum of reciprocals of divisors** of a perfect number equals 2 (harmonic mean of divisors is an integer — a lattice-like symmetry). The exponent \(p\) itself must be prime, and \(2^p - 1\) prime implies \(p\) prime — a **prime-chain constraint** analogous to root system rank restrictions in crystallogr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com