Euclid-Euler Theorem: Binary-Scaling Symmetry Links Even Perfect Numbers to Mersenne Primes — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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 = 2p−1(2p − 1) where 2p − 1 is prime (Mersenne prime). Equivalently, σ(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 2p−1(2p − 1) exhibits a **geometric ratio** of 2p − 1 to 2p−1, which for large p approaches 2 — but more strikingly, the **fractional part** of N/(2p − 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 2p − 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
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BibTeX
@misc{caldin2026euclideuler,
title = {{Euclid-Euler Theorem: Binary-Scaling Symmetry Links Even Perfect Numbers to Mersenne Primes — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658255},
url = {https://zenodo.org/records/23658255},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euclid-Euler Theorem: Binary-Scaling Symmetry Links Even Perfect Numbers to Mersenne Primes — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658255
UR - https://zenodo.org/records/23658255
AB - 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, σ(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
ER -