论文标题
与解决方案数量相关的三角形序列的Frobenius编号
The Frobenius number for sequences of triangular numbers associated with number of solutions
论文作者
论文摘要
Frobenius的著名线性双苯胺问题是确定最大整数（Frobenius编号）的问题，其表示$ a_1，\ dots，a_k $的表示数量最多为零，这是不可代表的。换句话说，所有大于此数字的整数至少可以用一种方式表示。该问题的自然概括之一是找到最大的整数（广义弗罗贝尼乌斯编号），其表示的数量最多是给定的非负整数$ p $。在两个变量的情况下，很容易找到此数字的明确形式。但是，即使在任何三个变量的特殊情况下，也都不知道明确的形式。在本文中，我们成功地展示了三角形数字三元组的广义弗罗贝尼乌斯数字的明确形式。当$ p = 0 $时，他们的Frobenius号码由Robles-Pérez和Rosales在2018年给出。
The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer $p$. It is easy to find the explicit form of this number in the case of two variables. However, no explicit form has been known even in any special case of three variables. In this paper we are successful to show explicit forms of the generalized Frobenius numbers of the triples of triangular numbers. When $p=0$, their Frobenius number is given by Robles-Pérez and Rosales in 2018.
