On the structure ofp-zero-sum free sequences and its application to a variant of Erdös-Ginzburg-Ziv theorem
W D Gao A Panigrahi R Thangadurai
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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 ℤ_{p} such that there is no subsequence of lengthp of S whose sum is zero in ℤ_{p}. Then we prove that we can arrange the sequence S as follows: $$S = (\underbrace {a,a,...,a,}_{u times}\underbrace {b,b,...,b,}_{v times}a'_1 ,a'_2 ,...,a'_{2p - k - u - v} )$$whereu ≥v,u +v ≥ 2p - 2k + 2 anda -b generates ℤ_{p}. 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 ℤ_{p} 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 ℤ_{p}.
W D Gao^{1} ^{} A Panigrahi^{1} ^{2} ^{} R Thangadurai^{1} ^{2} ^{}
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Volume 129 | Issue 5
November 2019
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