Research project Numerical semigroups are special sets of integers with intriguing properties and connections to number theory and combinatorics. This research explores how many such sets exist for a given number of gaps (genus) and how their structure behaves as the genus grows. The goal is to develop algorithms and compute semigroups of larger genus to uncover patterns that may impact cryptography, optimization, and advanced mathematics.
We are trying to determine how many numerical semigroups exist for a given genus. This is harder than it sounds when the number grows rapidly, and so far, researchers have only managed to compute up to genus 75. There are also fascinating patterns: growth appears to follow the golden ratio—the same proportion found in nature, art, and architecture! To push further, we are developing new algorithms and using powerful computers too.
A numerical semigroup is a subset S of N that contains 0, is closed under addition, and has a finite complement in N0. The elements in N0 minus S are called the gaps of the numerical semigroup, the largest gap is called the Frobenius number, and the number of gaps is the genus g(S) of the numerical semigroup.
There have been many efforts to compute the sequence counting the number of numerical semigroups of genus g, and today the sequence values are known up to n75. (For the complete list and more information, please click here.)
It was conjectured in 2007 that the sequence is increasing, that each term is at least the sum of the two previous terms, and that the ratio between each term and the sum of the two previous terms approaches one as the genus grows to infinity, which is equivalent to a growth rate approaching the golden ratio. The last statement of the conjecture was proved by Alex Zhai.
The first nonzero non-gap of a numerical semigroup is called the multiplicity of the semigroup, and it is proved to be asymptotically close to g(S) times (5+sqrt(5))/10 as g(S) grows to infinity. The i-th jump is the difference between the i-th and the (i-1)-th smallest non-gaps. This way, the multiplicity is the first jump.
We have developed a new algorithm to explore and count the numerical semigroups of a given genus which uses the unleaved version of the tree of numerical semigroups. In this project we plan to compute the number of numerical semigroups of genus larger than 70. We will study the asymptotic behavior of the first jumps. For this we need to make use of computational resources to compute the required data and the corresponding statistics.
