Frobenius number formula in terms of minimal generators

Establish a formula for the Frobenius number of a numerical semigroup in terms of the elements of its unique minimal set of generators.

Background

For a numerical semigroup, the Frobenius number is its largest gap, while the minimal generators are the irreducible elements that generate the semigroup. The paper identifies the classical Frobenius problem as an unresolved problem asking for a general formula that expresses the Frobenius number directly from these generators. The paper studies counting numerical semigroups by Frobenius number and by maximum primitive, but does not resolve this formula problem.

References

The famous open problem posed by Frobenius in the 1880s, known eponymously as the Frobenius problem or Frobenius' coin-exchange problem, asks to give a formula for the Frobenius number \Frobeniusoper(S) in terms of elements of the set \primitivesoper(S) .

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

At present, this problem is still open for numerical semigroups with embedding dimension greater or equal to three.

— Internal numerical semigroups  (2608.19984 - Casas et al., 20 Aug 2026) in Section 1, Introduction