What question did this study set out to answer?

The study aims to explore the relationship between the polynomial n(k) and its implications for modular forms and congruences.

July 6, 2026Open Access

Connection between the SU(n) Hierarchy and Modular Forms: Modular Properties of the Exact Polynomial n(k)

Key Points

The study aims to explore the relationship between the polynomial n(k) and its implications for modular forms and congruences.
Analyzed the polynomial n(k) = (103k⁴ − 370k³ + 101k² + 478k)/12 and its properties for various values of k.
Examined the periodicity of n(k) mod 12 and its connection to the weight of modular forms.
Studied occurrences of specific primes within n(k) and their relevance to Ramanujan congruences.
The values of n(k) for k = -1, 0, 1, 2, 3 yield denominators that divide n(1) = 26.
The unique perfect square found among n(k) for k > 0 is n(2) = 4.
Connections to prime 691 appear in both n(-4) and ζ(12), emphasizing the essence of Ramanujan's work.

Abstract

We investigate the connection between the exact polynomial n(k) = (103k⁴ − 370k³ + 101k² + 478k)/12, which generates the hierarchy of groups SU(n(k)), and modular forms and Ramanujan congruences. Key results:— The ratios n(k)/n(1) for k = -1, 0, 1, 2, 3 have denominators dividing n(1) = 26.— Periodicity n(k) mod 12 corresponds to the weight of the modular form Δ(τ).— n(2) = 4 is the unique perfect square among n(k) for k > 0.— The prime 691 appears in both n(-4) and ζ(12), connecting to the Ramanujan congruence.— All r₄(n(k)) are multiples of 12.— τ(n(k))/n(k) is an integer for k = -1.— n = 4 is the unique index for which σ₃(n) is prime. Languages: Russian, English. License: UAL v1.0.

Read Full Paperexternally

Demander à l'IA

Bookmark

View Full Paper