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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the solvability of the equation $ \sum\sp n\sb {i=1}x\sb i/d\sb i\equiv 0\;({\rm mod}\,1)$ and its application


Authors: Qi Sun and Da Qing Wan
Journal: Proc. Amer. Math. Soc. 100 (1987), 220-224
MSC: Primary 11D04
MathSciNet review: 884454
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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

$\displaystyle \frac{{{x_1}}}{{{d_1}}} + \frac{{{x_2}}}{{{d_2}}} + \cdots + \fra... ...),\quad {x_i}{\text{ integral}},{\text{1}} \leq {x_i} < {d_i}(1 \leq i \leq n),$

where $ {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} \cdots {d_n}/{d_i}) = 1$, or

2. If $ {d_{{i_1}}}, \ldots ,{d_{{i_k}}}(1 \leq i < \cdots < {i_k} \leq n)$ is the set of all even integers among $ \left\{ {{d_1}, \ldots ,{d_n}} \right\}$, then $ 2\nmid k,{d_{{i_1}}}/2, \ldots ,{d_{{i_k}}}/2$ are pairwise prime, and $ {d_{{i_j}}}$ is prime to any odd number in $ \{ {d_1}, \ldots ,{d_n}\} (j = 1, \ldots ,k)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0884454-6
PII: S 0002-9939(1987)0884454-6
Article copyright: © Copyright 1987 American Mathematical Society