论文标题
$ p $ - 数字的算术进展三倍
The $p$-numerical semigroup of the triple of arithmetic progressions
论文作者
论文摘要
对于给定的积极整数$ a_1，a_2，\ dots，a_k $ at $ \ gcd（a_1，a_2，a_2，\ dots，a_k）= 1 $，denumerant $ d（n）= d（n; a_1，a_1，a_2，a_2，a_2，\ dots，a_k）$ nonnon nornygitions $（x__1）线性方程$ a_1 x_1+a_2 x_2+\ dots+a_k x_k = n $用于正整数$ n $。对于给定的非负整数$ p $，令$ s_p = s_p（a_1，a_2，\ dots，a_k）$是所有非负整数$ n $的集合，以便$ d（n）> p $。在本文中，我们对$ P $ -FROBENIUS编号感兴趣，这是GAPS $ \ MATHBB N_0 \ BACKSLASH S_P $的最大值。这里$ \ mathbb n_0 $表示非负整数集。当$ p = 0 $时，$ s = s_0 $是原始数值半群，而$ 0 $ -FROBENIUS号是原始的Frobenius编号。两个变量的显式公式不仅以$ p = 0 $而闻名，还以$ p> 0 $而闻名，而且当有三个或更多变量时，即使在$ p = 0 $的特殊情况下也很难。对于$ p> 0 $，不仅更加困难，而且没有发现明确的公式。在本文中，$ p $ -FROBENIUS编号的明确公式和相关值是为算术进程的三倍。主要工具是确定$ p $-apéry集的元素。
For given positive integers $a_1,a_2,\dots,a_k$ with $\gcd(a_1,a_2,\dots,a_k)=1$, the denumerant $d(n)=d(n;a_1,a_2,\dots,a_k)$ is the number of nonnegative solutions $(x_1,x_2,\dots,x_k)$ of the linear equation $a_1 x_1+a_2 x_2+\dots+a_k x_k=n$ for a positive integer $n$. For a given nonnegative integer $p$, let $S_p=S_p(a_1,a_2,\dots,a_k)$ be the set of all nonnegative integers $n$'s such that $d(n)>p$. In this paper, we are interested in the $p$-Frobenius number, which is the maximum of the set of gaps $\mathbb N_0\backslash S_p$. Here $\mathbb N_0$ denotes the set of nonnegative integers. When $p=0$, $S=S_0$ is the original numerical semigroup, and the $0$-Frobenius number is the original Frobenius number. The explicit formula for two variables is known not only for $p=0$ but also for $p>0$, but when there are three or more variables, it is difficult even in the special case of $p=0$. For $p>0$, it is not only more difficult, but no explicit formula had been found. In this paper, explicit formulas of the $p$-Frobenius number and related values are given for the triple of arithmetic progressions. The main tool is to determine the elements of the $p$-Apéry set.
