On the structure ofp-zero-sum free sequences and its application to a variant of Erdös-Ginzburg-Ziv theorem



Letp be any odd prime number. Letk be any positive integer such that \(2 \leqslant k \leqslant \left[ {\frac{{p + 1}}{3}} \right] + 1\). LetS = (a 1,a 2,...,a 2p−k ) be any sequence in ℤp such that there is no subsequence of lengthp of S whose sum is zero in ℤp. Then we prove that we can arrange the sequence S as follows:
$$ S = (\underbrace {a,a,...,a,}_{u times}\underbrace {b,b,...,b,}_{v times}a'_1 ,a'_2 ,...,a'_{2p - k - u - v} ) $$
whereuv,u +v ≥ 2p - 2k + 2 anda -b generates ℤp. This extends a result in [13] to all primesp andk satisfying (p + 1)/4 + 3 ≤k ≤ (p + 1)/3 + 1. Also, we prove that ifg denotes the number of distinct residue classes modulop appearing in the sequenceS in ℤp of length 2p -k (2≤k ≤ [(p + 1)/4]+1), and \(g \geqslant 2\sqrt 2 \sqrt {k - 2} \), then there exists a subsequence of S of lengthp whose sum is zero in ℤp.


Sequences zero-sum problems zero-free Erdös-Ginzburg-Ziv theorem 


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Copyright information

© Indian Academy of Sciences 2005

Authors and Affiliations

  1. 1.Department of Computer Science and TechnologyUniversity of PetroleumBeijingChina
  2. 2.School of MathematicsHarish-Chandra Research InstituteAllahabadIndia

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