Mersenne Primes and A2 Lattices: A Search Result Gap Analysis — E8 Intelligence Research

FINDING: The search results are a scattered collection of number-theory pedagogy (primitive roots, Lucas-Lehmer test, Dirichlet spirals, unrefinable partitions of triangular numbers) with no direct, novel research linking Mersenne primes to A2 root systems or lattice points. The only "lattice" item is a lattice QCD paper on heavy-quark masses, which is unrelated to number theory. No new equation, constant, or ratio emerges from these sources. MATH: No new mathematics extracted. Known facts referenced: - Mersenne primes: \(M_p = 2^p - 1\), \(p\) prime. - Lucas–Lehmer test: \(s_0 = 4\), \(s_{k+1} = s_k^2 - 2 \pmod{M_p}\); \(M_p\) prime iff \(s_{p-2} \equiv 0\). - Primitive roots: \(g\) mod \(n\) has order \(\varphi(n)\). - Triangular numbers: \(T_n = n(n+1)/2\). - Dirichlet spirals: primes in arithmetic progressions \(a \bmod q\) with \(\gcd(a,q)=1\) distribute uniformly (Dirichlet's theorem), but the spiral pattern is a visualization artifact, not a new constant. CONNECTION: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com