Congruence Families for 4- and 6-Regular Partitions — E8 Intelligence Research

FINDING: The search results are largely pedagogical (introductory congruence lectures) and one recent arXiv paper on Ramanujan-type congruences for 4,6-regular partitions — no direct McKay-Thompson series material surfaced. The core mathematical content is the existence of infinite families of congruences for partition functions \( b_4(n) \) and \( b_6(n) \) modulo primes \( m \), with explicit examples. MATH: - For \( b_4(n) \) (4-regular partitions): congruences exist modulo primes \( m ≥ 2 \), infinitely many families. - For \( b_6(n) \) (6-regular partitions): congruences modulo primes \( m ≥ 5 \), infinitely many families. - The generating functions are modular forms of half-integral weight; congruences arise from the action of Hecke operators on spaces of cusp forms. - Explicit examples (from the paper, not shown in the search snippet) would involve specific primes \( m \) and arithmetic progressions \( an+b \) such that \( b_k(an+b) ≡ 0 {m} \). - No co Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com