Wilf’s conjecture for all numerical semigroups

Prove or disprove Wilf’s conjecture by establishing whether every numerical semigroup S satisfies the inequality (F(S) + 1 − g(S)) / (F(S) + 1) ≥ 1 / e(S), where F(S) is the Frobenius number, g(S) is the genus, and e(S) is the embedding dimension.

Background

The paper reviews core invariants of numerical semigroups—embedding dimension, Frobenius number, and genus—and recalls Wilf’s conjecture, which ties these three parameters via a density inequality. Despite substantial progress on related bounds and random models, Wilf’s conjecture remains a central open problem in the field.

The authors’ work focuses on probabilistic bounds in Erdős–Rényi-type random numerical semigroup models; however, Wilf’s conjecture concerns all numerical semigroups and stands independently of the random model considered here.

References

A central problem in numerical semigroups is to prove or disprove Wilf's conjecture which is that every numerical semigroup $S$ satisfies the inequality \frac{\F(S) + 1 - \g(S)}{\F(S)+1} \geq \frac{1}{\e(S)} .

— Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups  (2411.13767 - Bogart et al., 2024) in Section 1 (Introduction)

Another open problem which has received considerable attention is a question of Wilf, now popularly known as Wilf's conjecture. For a numerical semigroup S, we define the set of left elements to be S \cap [0,\Frobeniusoper(S)) and denote it by \leftsoper(S). Wilf asked whether the following inequality always holds |\primitivesoper(S)|\cdot |\leftsoper(S)|\geq \Frobeniusoper(S)+1.

— On counting numerical semigroups by maximum primitive and Wilf's conjecture  (2501.04417 - Delgado et al., 8 Jan 2025) in Section 1, Introduction

A central problem in numerical semigroups is to prove or disprove Wilf's conjecture which is that every numerical semigroup $$ satisfies the inequality \frac{\F[] + 1 - \g()} {\F[]+1} \geq \frac{1}{\e()} involving all three parameters. That is, the reciprocal of the embedding dimension is conjectured to be a lower bound on the density of the segment of $$ between 0 and the Frobenius number.

— Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups  (2411.13767 - Bogart et al., 2024) in Section 1: Introduction