Euclid-Euler Theorem: Binary Scaling 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 structure in arithmetic perfection. | MATH: Even perfect number N = 2p−1(2p − 1) where 2p − 1 is prime (Mersenne prime). Equivalently, σ(N) = 2N (sum of divisors). The theorem: N is even perfect iff N = 2p−1Mp with Mp = 2p − 1 prime. | CONNECTION: The factor 2p−1 is a pure binary shift — a scaling symmetry in base-2. The Mersenne prime Mp = 2p − 1 is the largest power of 2 minus 1, linking to binary repunits. The ratio N / 2p−1 = Mp — 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 Dp 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
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BibTeX
@misc{caldin2026euclideuler,
title = {{Euclid-Euler Theorem: Binary Scaling Links Even Perfect Numbers to Mersenne Primes — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635327},
url = {https://zenodo.org/records/23635327},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euclid-Euler Theorem: Binary Scaling Links Even Perfect Numbers to Mersenne Primes — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635327
UR - https://zenodo.org/records/23635327
AB - 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 = 2^p−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 = 2^p−1M_p with M_p = 2^p − 1 prime. | CONNECTION: The factor 2^p−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 / 2^p−1 = M_p — the "defect" from the next power of 2. In geometric terms, 2^p−1 is the order of the hyperoctahedral group's binary subgroup; the structure echoes root lattice A_p−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 2^k(2^k+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
ER -