3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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 $(x_1,a_2)=1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.