## Weighted sums in finite cyclic groups.(English)Zbl 1052.11014

Let $$C_n$$ be the cyclic group of $$n$$ elements, $$p$$ be a prime, and $$k\in\mathbb{N}$$. Let $$\{w_1,\dots,w_k\}$$ be a sequence of $$k$$ integers such that $$w_1+\cdots+ w_k \equiv 0\pmod{p^2}$$. Then, for every sequence $$a_1,a_2,\dots$$, of $$p^2+1$$ elements in $$C_{p^2}$$, there are $$k$$ distinct indices $$i_1,i_2,\dots,i_k$$ such that $$w_1a_{i_1}+ \cdots+ a_{i_k}=0$$. This generalizes a result of Y. O. Hamidoune [Discrete Math. 162, No. 1–3, 127–132 (1996; Zbl 0872.11016).

### MSC:

 11B75 Other combinatorial number theory 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20K01 Finite abelian groups

### Keywords:

zero-sum; cyclic group; sequence

Zbl 0872.11016
Full Text:

### References:

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