Let $a_1,\ a_2,\ a_3,\ \ldots,\ a_n$ be distinct positive integers. Find $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$ such that: $$\left\{\begin{array}{rl}(x_1,x_2,\ldots,x_n)&=1\\ a_2x_1+a_3x_2+\cdots+a_nx_{n-1}+a_1 x_n & =y x_2\\ \cdots\cdots\cdots\cdots & = \cdots\cdots \\ a_n x_1+a_1 x_2+ \cdots + a_{n-1} x_n&=y x_n \end{array}\right.$$

How to solve this system equation ?

I am not good at diophantine equation and the problem above is so hard.

  • $\begingroup$ The first condition is redundant; if the $x_i$'s have a common factor we may divide through by that factor in all the other equations. $\endgroup$ – vadim123 May 21 '13 at 3:24

By inspection $x_1=x_2=\ldots=x_n=1,y=\sum a_i$ is a solution.

The system is underspecified, though, so there are other solutions, e.g. for $n=2$ $$ x_2=a_2, y=a_1+x_1 $$ is a solution for any choice of $x_1$ with $(x_1,a_2)=1$.


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