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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 Diophantine equation $ \sum\sp n\sb {i=1}x\sb i/d\sb i\equiv 0\pmod 1$


Authors: Qi Sun and Da Qing Wan
Journal: Proc. Amer. Math. Soc. 112 (1991), 25-29
MSC: Primary 11D04
MathSciNet review: 1047008
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Abstract: Let $ {d_1}, \ldots ,{d_n}$ be $ n$ positive integers. The purpose of this note is to study the number of solutions and the least solutions of the following diophantine equation:

$\displaystyle (1)\quad \frac{{{x_1}}}{{{d_1}}} + \cdots + \frac{{{x_n}}}{{{d_n}}} \equiv 0(\bmod 1),\quad 1 \leq {x_i} \leq {d_i} - 1,$

which arises from diagonal hypersurfaces over a finite field. In particular, we determine all the $ {d_i}$'s for which (1) has a unique solution.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1047008-8
PII: S 0002-9939(1991)1047008-8
Article copyright: © Copyright 1991 American Mathematical Society