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arXiv 2108.04756math.NT
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关于线性丢番图方程的整数非负解的个数

On the number of integer non-negative solutions of a linear Diophantine equation

Eteri Samsonadze

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摘要：

研究了方程$\sum_{i=1}^{n} a_i x_i=b$的整数非负解的个数$P(b)$，其中$a_1,a_2,...,a_n$为自然数，$b$为非负整数.与传统的研究$P(b)$函数的方法不同，我们的研究不使用数列理论的技巧，而是主要利用Kronecker函数的性质和组合学的元素。导出了表示$P(b)$的公式，对于整数非负$b$，通过$P(r)$，$P(r+M)$，.，当$s\neq 0$时，$P(r+(s-1)M)$，其中$s=\left[ n-\dfrac{\sum_{i=1}^{n}a_i+r}{M} \right]$和在某些特定情况下取相当小的值; $M$是数$a_1,a_2,\ldots,a_n$的最小公倍数，$r$是$b$模$M$的余数。对于任意非负整数$b$，导出了计算$P(b)$的递推公式，特别是用于求$P(r)$，$P(r+M)$，.，$P(r+(s-1)M)$。对于$s=0$和$a_1$，$a_2$，.，$a_n$是互质的，给出了$P(b)=\dfrac{M^{n-1}}{a_1a_2\ldots a_n}C^{n-1}_{\left[\dfrac{b}{M}\right]+n-1}$的显式表达式.为了说明所提出的方法，寻找函数$P(b)$的线性丢番图方程的$2,3,7$和$n$变量的例子。

英文摘要：

We deal with the problem to find the number $P(b)$ of integer non-negative solutions of an equation $\sum_{i=1}^{n} a_i x_i=b$, where $a_1,a_2,...,a_n$ are natural numbers and $b$ is a non-negative integer. As different from the traditional methods of investigation of the function $P(b)$, in our study we do not employ the techniques of number series theory, but use in the main the properties of the Kronecker function and the elements of combinatorics. The formula is derived to express $P(b)$, for an integer non-negative $b$, via $P(r)$, $P(r+M)$,..., $P(r+(s-1)M)$ when $s\neq 0$, where $s=\left[ n-\dfrac{\sum_{i=1}^{n}a_i+r}{M} \right]$ and takes quite small values in some particular cases; $M$ is the least common multiple of the numbers $a_1,a_2,\ldots,a_n$, and $r$ is the remainder of $b$ modulo $M$. Also, the recurrent formulas are derived to calculate $P(b)$, for any non-negative integer $b$, which, in particular, are used in finding $P(r)$, $P(r+M)$,..., $P(r+(s-1)M)$. For the case where $s=0$ and $a_1$, $a_2$,..., $a_n$ are coprime, the explicit formula $P(b)=\dfrac{M^{n-1}}{a_1a_2\ldots a_n}C^{n-1}_{\left[\dfrac{b}{M}\right]+n-1}$ is given. To illustrate the proposed method, examples of finding the function $P(b)$ for linear Diophantine equations with $2,3,7$ and $n$ variables are given.

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