8 Prime Atlas
GIFT achieves 100% coverage of primes \({\lt} 200\) through explicit expressions.
8.1 Tier 1: Direct Constants
Direct GIFT constants that are prime: \(\{ 2, 3, 5, 7, 11, 13, 17, 19, 31, 61\} \)
Every element of tier1_primes is prime.
8.2 Heegner Numbers
The 9 Heegner numbers are the only \(d\) such that \(\mathbb {Q}(\sqrt{-d})\) has class number 1.
\(\{ 1, 2, 3, 7, 11, 19, 43, 67, 163\} \)
\(163 = \mathrm{dim}(E_8) - \mathrm{rank}(E_8) - b_3= 248 - 8 - 77\)
All 9 Heegner numbers have GIFT expressions.