一类特殊不定方程解的问题

On a special of indeterminate equation

首先给出了下列不定方程ａ１ａ２…ａｋ＝Σｋｉ＝１ａｎｉ，其中，ａ１，ａ２，…，ａｋ∈Ｎ°＝｛０，１，２，３，４，５，６，７，８，９｝，ｋ∈Ｎ，Ｎ为自然数集，就此方程在Ｎ°上有关解的问题作者提出了如下两个问题：（１）此方程在Ｎ°上是否存在解？（２）若此方程有解，则解的个数为多少？其次，就此问题进行了一些讨论，对不同的自然数ｋ和ｎ。

In this paper, we first consider the following indeterminate equation a 1a 2…a k=Σki=1a n i Where a 1,a 2,…,a k∈N°={0,1,2,…9},a 1,a 2…a k represents a k-place natural number, n and k are two natural numbers. We propose the two following questions on the problem of equation of variabe a 1,a 2,…,a k: (1)Is there any solution to this equation in N°? (2)If there exist some solutions in N°,how many solutions are there? and then, we discuss this problem and obtain some special solutions for some different natural numbers k and n.