On the Diophantine equation ∑ⁿᵢ₌₁𝑥ᵢ/𝑑ᵢ≡0 (mod 1)

Sun, Qi; Wan, Da Qing

doi:10.1090/S0002-9939-1991-1047008-8

On the Diophantine equation $\sum ^ n_ {i=1}x_ i/d_ i\equiv 0\pmod 1$
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by Qi Sun and Da Qing Wan PDF

Proc. Amer. Math. Soc. 112 (1991), 25-29 Request permission

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