Mersenne Primes as Binary Repunits and Their Link to Perfect Numbers — E8 Intelligence Research

FINDING: Mersenne primes are binary repunits (all 1s in base 2), and every Mersenne prime is of the form 4n+3, linking them to modular arithmetic and perfect numbers. | MATH: Mersenne prime: M_p = 2^p − 1 (p prime). Binary repunit: M_p = (111...1)_2 with p ones. Congruence: M_p ≡ 3 (mod 4) for p ≥ 2, hence M_p = 4n+3. Perfect number: N = 2^(p−1)(2^p − 1) = 2^(p−1) · M_p. | CONNECTION: Binary repunit structure is a degenerate geometric series: M_p = 1 + 2 + 4 + ... + 2^(p−1) = (2^p − 1)/(2 − 1). The ratio of successive terms in this sum is exactly 2 — not the golden ratio, but the base-2 analogue of the golden ratio's self-similarity. The 4n+3 form aligns with the crystallographic restriction theorem: only certain modular residues are admissible for rotational symmetries in lattices (e.g., 3-fold, 4-fold, 6-fold — but 4n+3 primes like 7, 31, 127 are congruent to 3 mod 4, which is the signature of quaternion-like or non-commutative structures, not simple 2D lattices). | DEPTH: 6 — Solid Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com