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 Szemerédi 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 Szemerédi 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 Szemerédi. 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 the hypotheses are non-void is $C_2\oplus C_{2m}$.

格雷厄姆（Graham）的一个古老推测认为，如果 $ñ$ 是素数 $\mathrm{小号}$ 是一个序列 $ñ$ 循环群的术语 $C_{ñ}$ 这样所有（非平凡的）零和子序列都具有相同的长度，则 $\mathrm{小号}$最多包含两个不同的术语。1976年，Erdős和Szemerédi给出了关于足够大素数的猜想的证明$ñ$。但是，证明非常复杂，以至于无法确定小素数的细节。在Erdős和Szemerédi的论文中以及在Erdős和Graham的后来调查中，都为证明的复杂性感到遗憾。最近，一个新的证明即使对非素数也有效$ñ$是由高，哈米多恩和王给出的，使用的是Savchev和Chen最近证明的结构定理，它针对零长度的零和自由序列。 $C_{ñ}$。但是，由于这是一个相当复杂的结果，因此他们并不认为这是Erdős，Graham和Szemerédi寻求的简单证明。在本文中，我们给出了仅使用柯西-达文波特定理和信鸽原理的原始猜想的简短证明，因此也许有资格作为简单证明。用Devos-Goddyn-Mohar定理代替Cauchy-Davenport定理的使用，我们获得了非素数情况的替代证明，尽管不是那么简单。此外，我们的方法还产生了详尽的清单，详细列出了$\mathrm{小号}$ 并且适用于任意有限的阿贝尔群，尽管唯一的假设是非无效的非循环群是 $C_{\mathrm{2个}}\oplus C_{\mathrm{2个}米}$。