arXiv 0903.32002009-03-18math.COmath.NT

关于Graham一个猜想的注记

Note on a Conjecture of Graham

David J. Grynkiewicz

摘要：

格雷厄姆的一个旧猜想指出，如果$n$是一个素数，并且$S$是循环群$C_n$中的$n$项的序列，使得所有（非平凡的）零和序列具有相同的长度，则$S$必须包含最多两个不同的项。1976年，Erdés和Szemeredi给出了充分大素数$n$猜想的证明。然而，证明是足够复杂的小素数的细节从来没有制定出来。无论是在埃尔德什和Szemeredi的文件，并在后来的调查埃尔德什和格雷厄姆，复杂的证明感到遗憾。最近，Gao，Hamidoune和Wang利用Savchev和Chen最近证明的$C_n$中长长度零和自由序列的结构定理，给出了一个新的证明，甚至对非素数$n$也有效。然而，因为这是一个相当复杂的结果，他们不相信这是简单的证明所寻求的Erdjs，格雷厄姆和Szemeredi。本文仅利用柯西-达文波特定理和鸽子洞原理对原猜想给出了一个简短的证明，因此可能是一个简单的证明。用Devos-Goddyn-Mohar定理代替Cauchy-Davenport定理，我们得到了非素情形的另一种证明，尽管不那么简单。此外，我们的方法产生了一个详尽的列表，详细说明了$S$的精确结构，并适用于任意有限阿贝尔群，虽然唯一的非循环群，这是非平凡的是$C_2\oplus C_{2m}$。

英文摘要：

An old conjecture of Graham stated that if $n$ is a prime and $S$ is a sequence of $n$ terms from the cyclic group $C_n$ such that all (nontrivial) zero-sum subsequences have the same length, then $S$ must contain at most two distinct terms. In 1976, Erdős and Szemeredi gave a proof of the conjecture for sufficiently large primes $n$. However, the proof was complicated enough that the details for small primes were never worked out. Both in the paper of Erdős and Szemeredi and in a later survey by Erdős and Graham, the complexity of the proof was lamented. Recently, a new proof, valid even for non-primes $n$, was given by Gao, Hamidoune and Wang, using Savchev and Chen's recently proved structure theorem for zero-sum free sequences of long length in $C_n$. However, as this is a fairly involved result, they did not believe it to be the simple proof sought by Erdős, Graham and Szemeredi. In this paper, we give a short proof of the original conjecture that uses only the Cauchy-Davenport Theorem and pigeonhole principle, thus perhaps qualifying as a simple proof. Replacing the use of the Cauchy-Davenport Theorem with the Devos-Goddyn-Mohar Theorem, we obtain an alternate proof, albeit not as simple, of the non-prime case. Additionally, our method yields an exhaustive list detailing the precise structure of $S$ and works for an arbitrary finite abelian group, though the only non-cyclic group for which this is nontrivial is $C_2\oplus C_{2m}$.

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英文摘要：

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