In this paper, we shall address three closely-related conjectures due to van Emde Boas, W D Gao and Kemnitz on zero-sum problems on Z_p ⊗ Z_p. We prove a number of results including a proof of the conjecture of Gao for the primep = 7 (Theorem 3.1). The conjecture of Kemnitz is also proved (Propositions 4.6, 4.9, 4.10) for many classes of sequences.

In this note, we generalize some theorems on zero-sums with weights from [1], [4] and [5] in two directions. In particular, we consider $\mathbb{Z}^d_p$ for a general 𝑑 and subgroups of $Z^∗_p$ as weights.

Erdős and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation $x(x+1)(x+2)\ldots(x+(m-1))=y^n$ has no solutions in positive integers $x,m,n$ where $m,n>1$ and $y\in Q$. We consider the equation

$$(x-a_1)(x-a_2)\ldots(x-a_k)+r=y^n$$

where $0\leq a_1 < a_2 <\cdots < a_k$ are integers and, with $r\in Q,n\geq 3$ and we prove a finiteness theorem for the number of solutions 𝑥 in $Z,y$ in 𝑄. Following that, we show that, more interestingly, for every nonzero integer $n>2$ and for any nonzero integer 𝑟 which is not a perfect 𝑛-th power for which the equation admits solutions, 𝑘 is bounded by an effective bound.