Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The first elements of the quotient of a numerical semigroup by a
  positive integer   Given three pairwise coprime positive integers $a_1,a_2,a_3 \in \mathbb{Z}^+$
we show the existence of a relation between the sets of the first elements of
the three quotients $\frac{\langle a_i,a_j \rangle}{a_k}$ that can be made for
every $\{i.j,k\}=\{1,2,3\}$. Then we use this result to give an improved
version of Johnson's semi-explicit formula for the Frobenius number
$g(a_1,a_2,a_3)$ without restriction on the choice of $a_1,a_2,a_3$ and to give
an explicit formula for a particular class of numerical semigroups.

27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th