## The Erdős-Ginzberg-Ziv theorem with units.(English)Zbl 1206.11032

Author’s summary: “Let $$x_1,\dots ,x_r$$ be a sequence of elements of $$\mathbb Z_n$$, the integers modulo $$n$$. How large must $$r$$ be to guarantee the existence of a subsequence $$x_{i_1},\dots ,x_{i_n}$$ and units $$\alpha_1,\dots ,\alpha_n$$ with $$\alpha_1x_{i_1}+\dots +\alpha_nx_{i_n}=0$$? The author’s main aim in this paper is to show that $$r=n+a$$ is large enough, where $$a$$ is the sum of the exponents of primes in the prime factorization of $$n$$. This result, which is best possible, could be viewed as a unit version of the Erdős-Ginzberg-Ziv theorem. This proves a conjecture of S. D. Adhikari, Y. G. Chen, J. B. Friedlander, S. V. Konyagin, and F. Pappalardi [Discrete Math. 306, No. 1, 1–10 (2006; Zbl 1161.11311)].
He also discusses a number of related questions, and make conjectures which would greatly extend a theorem of Gao.”
Reviewer’s remark: The main result of the paper was also proved independently by F. Luca [Discrete Math. 307, No. 13, 1672–1678 (2007; Zbl 1123.11012)] using a somewhat different method.

### MSC:

 11B50 Sequences (mod $$m$$) 11B75 Other combinatorial number theory

### Keywords:

zero-sum problems; Erdős-Ginzberg-Ziv theorem

### Citations:

Zbl 1123.11012; Zbl 1161.11311
Full Text:

### References:

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