Heegner Numbers and Integer j-Invariants: Class Number One and Ramanujan Series — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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) Σ_{n=0}^∞ ( (6n)! / (n!)³ ) · (545140134n + 13591409) / (640320³)^{n+1/2}. Moment asymptotics: Σ_{d>0, ε_d ≤ x} h(d)^k ~ C_k x^{2k+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
Related
- E8 Heegner‑Phi Resonance: Number‑Theoretic Control of Golden‑Angle Quantum Coherence — E8 Intelligence Research
- E8 Phi‑Twisted Orbit Spectra Yield New Ramanujan‑Type Series for 1/π — E8 Intelligence Research
- Rogers-Ramanujan Resonance Modes in E8 Root Vector Lattice — E8 Intelligence Research
- The Golden Ratio Bridge: Rogers-Ramanujan Continued Fraction and Quintic Solvability — E8 Intelligence Research
- MERLIN SCIENCE — Ramanujan's Tau Function and the Leech Lattice: A Deep Arithmetic Link — E8 Intelligence Research
BibTeX
@misc{caldin2026heegner,
title = {{Heegner Numbers and Integer j-Invariants: Class Number One and Ramanujan Series — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23616838},
url = {https://zenodo.org/records/23616838},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Heegner Numbers and Integer j-Invariants: Class Number One and Ramanujan Series — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23616838
UR - https://zenodo.org/records/23616838
AB - 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) Σ_{n=0}^∞ ( (6n)! / (n!)³ ) · (545140134n + 13591409) / (640320³)^{n+1/2}. Moment asymptotics: Σ_{d>0, ε_d ≤ x} h(d)^k ~ C_k x^{2k+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
ER -