Frobenius Problem on Four Elements and Symmetry of Semigroups

令 $a,b,c,d$ 為一組獨立的正整數。若一個非負整數可表為 $c_1a+c_2b+c_3c+c_4d$ 的形式，其中 $c_i$ 均為非負整數，則稱它可被 $a,b,c,d$ 表示。

我們將給出在特殊情形中，不能由 $a,b,c,d$ 表出的非負整數個數 $n(a,b,c,d)$，及最大不可表的整數 $g(a,b,c,d)$。最後並討論由 $a,b,c,d$ 生成的半群對稱性。

Let $a,b,c,d$ be independent positive integers.
A nonnegative integer is said to be represented by $a,b,c,d$ if it can be represented as the form $c_1a+c_2b+c_3c+c_4d$, where the $c_i$'s are nonnegative integers.

We will find the number $n(a,b,c,d)$ of nonnegative integers cannot be represented by $a,b,c,d$, and the number $g(a,b,c,d)$ which is the largest integer cannot be represented by $a,b,c,d$ in some special cases. Finally we discuss the symmetry property of the semigroup generated by $a,b,c,d$.