Betti counts of five-generator numerical semigroup families

How large can the first Betti number, the minimal toric-ideal generator count, be for a numerical semigroup with five minimal generators? The recorded family is generated by m,m+1,m+k,m+2k−1,m+2k with m=2kq+1, k≥3 and q≥k−1.

Why it matters and what counts as progress

Why it matters: An exact unbounded five-generator family quantifies growth of the relation complexity of numerical-semigroup rings.

What counts as progress: Establish the exact 4k+2 generator count and a nonredundant presentation, or find new growth bounds for five-generated semigroups.

How the problem developed

2023 · Partial result [1]
Giulio Caviglia, Alessio Moscariello and Alessio Sammartano Bounded semigroup-ring Betti numbers in terms of generator width and proved the Herzog–Stamate bound for four generators outside finitely many widths.
18 Oct 2026 · Reported claim [2]
Ben Keene and GPT-6 Astra Reported an explicit five-generated family with generators m,m+1,m+k,m+2k−1,m+2k, where m=2kq+1, whose toric ideal has exactly 4k+2 minimal generators for k≥3 and q≥k−1.

AI contribution

Astra finds a five-generator semigroup family with exact Betti count

For k≥3 and q≥k−1, let m=2kq+1. The numerical semigroup minimally generated by m,m+1,m+k,m+2k−1,m+2k has toric-ideal generator count 4k+2. The formal development also gives a nonredundant presentation.

AI involvementAI-centered

ImportanceModest

Technical verificationFormal (Lean 4 kernel, NanoDa and con-ron through Comparator)

Human understandingFew