Heegner Numbers and Integer j-Invariants: Class Number One and Ramanujan Series — E8 Intelligence Research

FINDING: Heegner numbers (d = −1, −2, −3, −7, −11, −19, −43, −67, −163) are the only negative discriminants with class number h(−d) = 1, yielding integer j-invariants at imaginary quadratic points; recent work extends moments of class numbers for indefinite forms and Ramanujan-type series tied to d = 163. | MATH: Class number h(−d)=1 for d ∈ {1,2,3,7,11,19,43,67,163}. j((1+√−163)/2) = −640320³ = −262537412640768000 (integer). Ramanujan series: 1/π = (√163/8) Σₙ₌₀^∞ ( (6n)! / (n!)³ ) · (545140134n + 13591409) / (640320³)n+1/2. Moment asymptotics: Σd>0, ε_d ≤ x h(d)^k ~ C_k x²ᵏ⁺¹ (log x)−? (per arXiv:1609.01630). | CONNECTION: 163 ≈ 0.618 × 264 (φ-linked? No — but 163 = 4×40.75, not golden). However, 640320³ = (2⁶·3·5·23·29)³ — factors align with 2,3,5 (base-60 primes). j-invariant values at Heegner discriminants are algebraic integers; for d = −163, the j-value is an integer whose prime factorization includes 2⁶·3³·5³·23³·29³ — all primes ≤ 29, echoing crystallographic ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com