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.

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 = (a₁,a₂,...,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.

Let 𝑝 be a prime number. Let 𝐺 be a finite abelian 𝑝-group of exponent 𝑛 (written additively) and 𝐴 be a non-empty subset of $]n[:=\{1,2,\ldots,n\}$ such that elements of 𝐴 are incongruent modulo 𝑝 and non-zero modulo 𝑝. Let $k \geq D(G)/|A|$ be any integer where $D(G)$ denotes the well-known Davenport’s constant. In this article, we prove that for any sequence $g_1,g_2,\ldots,g_k$ (not necessarily distinct) in 𝐺, one can always extract a subsequence $g_{i_1},g_{i_2},\ldots,g_{i_l}$ with $1 \leq l \leq k$ such that

$$\sum\limits_{j=1}^l a_j g_{i_j}=0 \text{in} G,$$

where $a_j\in A$ for all 𝑗. We provide examples where this bound cannot be improved. Furthermore, for the cyclic groups, we prove some sharp results in this direction. In the last section, we explore the relation between this problem and a similar problem with prescribed length. The proof of Theorem 1 uses group-algebra techniques, while for the other theorems, we use elementary number theory techniques.

In 1935, Erdös proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.

Given an integer $N\geq 3$, we shall prove that for all primes $p\geq(N-2)^2 4^N$, there exists 𝑥 in $(\mathbb{Z}/p\mathbb{Z})^∗$ such that $x,x+1,\ldots,x+N-1$ are all squares (respectively, non-squares) modulo 𝑝. Similarly, for an integer $N\geq 2$, we prove that for all primes $p\geq \exp(2^{5.54N})$, there exists an element $x\in(\mathbb{Z}/p\mathbb{Z})^∗$ such that $x,x+1,\ldots,x+N-1$ are all generators of $(\mathbb{Z}/p\mathbb{Z})^∗$.

In this article, we prove that infinite number of integers satsify Erdős–Woods conjecture. Moreover, it follows that the number of natural numbers $\leq x$ satisfies Erdős–Woods conjecture with 𝑘 = 2 is at least 𝑐𝑥/(log 𝑥) for some positive constant 𝑐 > 2.