Articles written in Proceedings – Mathematical Sciences

• Gao’s conjecture on zerosum sequences

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 Zp ⊗ Zp. 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.

• On the structure ofp-zero-sum free sequences and its application to a variant of Erdös-Ginzburg-Ziv theorem

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,...,a2p−k) be any sequence in ℤp such that there is no subsequence of lengthp of S whose sum is zero in ℤp. Then we prove that we can arrange the sequence S as follows: $$S = (\underbrace {a,a,...,a,}_{u times}\underbrace {b,b,...,b,}_{v times}a'_1 ,a'_2 ,...,a'_{2p - k - u - v} )$$whereuv,u +v ≥ 2p - 2k + 2 anda -b generates ℤp. This extends a result in  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 ℤp 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 ℤp.

• A Variant of Davenport's Constant

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.

• On a Paper of S S Pillai

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.

• Distribution of Residues and Primitive Roots

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 $(\mathbb{Z}/p\mathbb{Z})^∗$.

• On Erdős–Wood’s conjecture

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 &gt; 2.

• Editorial Note on Continuous Article Publication

Posted on July 25, 2019

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