The Minimum Isolated Gap in Numerical Semigroups Generated by two Elements

Abstract

A numerical semigroup is a cofinite additive submonoid of the non-negative integers. The study of its gaps has long been central to the theory, with particular emphasis on invariants such as the Frobenius number and the set of pseudo-Frobenius numbers. In this setting, an \emph{isolated gap} is a gap $g$ such that both $g-1$ and $g+1$ belong to the semigroup, and a numerical semigroup is called \emph{perfect} if it has no isolated gaps. Motivated by Smith’s construction of perfect numerical semigroups of embedding dimension three via adjoining the minimum isolated gap of a two-generated semigroup, we derive explicit formulas for the minimum isolated gap of the two-generated numerical semigroups $\langle a, a+n \rangle$ for $n=2,3,4,5,6$, under the natural condition $\gcd(a,a+n)=1$ together with the relevant congruence restrictions on $a$. Our approach combines congruence arguments based on the Euclidean algorithm and the Chinese Remainder Theorem with elementary computations. As a consequence, we obtain constructive families of perfect numerical semigroups of embedding dimension three obtained by adjoining these minimum isolated gaps.