Euler's Totient Function: Multiplicativity, Even Image, and Missing Even Numbers — E8 Intelligence Research

FINDING: Euler's totient function φ (n) counts integers 1 ≤ k ≤ n coprime to n; it is multiplicative for coprime factors, and its image consists of 1 and even numbers only, with many even numbers missing. MATH: - φ (n) = n ∏₏|₍ (1 − 1/p) - φ (mn) = φ (m) φ (n) if gcd (m, n) =1 - φ (n) is even for n > 2 - Euler's theorem: a^φ (n) ≡ 1 (mod n) for gcd (a, n) =1 - Fermat's little theorem: a^ (p−1) ≡ 1 (mod p) for prime p ∤ a CONNECTION: - φ (n) even for n>2 implies a parity constraint reminiscent of crystallographic restriction: only rotations of order 2, 3, 4, 6 are allowed in 2D/3D lattices. The evenness of φ (n) for n>2 mirrors the fact that only n with φ (n) even (i. e. , n≠1, 2) can appear as orders of symmetry in certain modular contexts. - No direct golden ratio or base-60 link in these results; the multiplicative structure of φ (n) relates to prime factorisation, not to harmonic ratios. DEPTH: 4 — Foundational number theory result with cryptographic applications, but no direct geom Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com