On the solvability of the equation ∑ⁿᵢ₌₁𝑥ᵢ/𝑑ᵢ≡0(𝑚𝑜𝑑1) and its application

Abstract

In this paper, we obtain a necessary and sufficient condition under which the equation of the title is unsolvable. More precisely, for the equation \[ x 1 d 1 + x 2 d 2 + ⋯ + x n d n ≡ 0 ( mod 1 ) , x i integral , 1 ≤ x i > d i ( 1 ≤ i ≤ n ) , \frac {{{x_1}}}{{{d_1}}} + \frac {{{x_2}}}{{{d_2}}} + \cdots + \frac {{{x_n}}}{{{d_n}}} \equiv 0\quad (\bmod 1),\quad {x_i}{\text { integral}},{\text {1}} \leq {x_i} > {d_i}(1 \leq i \leq n), \] where d 1 , … , d n {d_1}, \ldots ,{d_n} are fixed positive integers, we prove the following result: The above equation is unsolvable if and only if 1. For some d i , ( d i , d 1 d 2 ⋯ d n / d i ) = 1 {d_i},({d_i},{d_1}{d_2} \cdots {d_n}/{d_i}) = 1 , or 2. If d i 1 , … , d i k ( 1 ≤ i > ⋯ > i k ≤ n ) {d_{{i_1}}}, \ldots ,{d_{{i_k}}}(1 \leq i > \cdots > {i_k} \leq n) is the set of all even integers among { d 1 , … , d n } \left \{ {{d_1}, \ldots ,{d_n}} \right \} , then 2 ∤ k , d i 1 / 2 , … , d i k / 2 2\nmid k,{d_{{i_1}}}/2, \ldots ,{d_{{i_k}}}/2 are pairwise prime, and d i j {d_{{i_j}}} is prime to any odd number in { d 1 , … , d n } ( j = 1 , … , k ) \{ {d_1}, \ldots ,{d_n}\} (j = 1, \ldots ,k) .