R. Thangadurai

A variant of Kemnitz conjecture

For any integer n ≥ 3, by g(Zn ⊕ Zn) we denote the smallest positive integer t such that every subset of cardinality t of the group Zn ⊕ Zn contains a subset of cardinality n whose sum is zero. Kemnitz (Extremalprobleme fur Gitterpunkte, Ph.D. Thesis, Technische Universitat Braunschweig, 1982) proved that g(Zp ⊕ Zp) = 2p - 1 for p = 3, 5, 7. In this paper, as our main result, we prove that g(Zp ⊕ Zp) = 2p - 1 for all primes p ≥ 67.

Olson's constant for the group Z p O Z p

Let G be a finite abelian group. By Ol(G), we mean the smallest integer t such that every subset A ⊂ G of cardinality t contains a non-empty subset whose sum is zero. In this article, we shall prove that for all primes p > 4.67 × 1034, we have Ol(Zp ⊕ Zp) = p + Ol(Zp) - 1 and hence we have Ol(Zp ⊕ Zp) ≤ p - 1 + ⌈ √2p + 5 log p⌉. This, in particular, proves that a conjecture of Erdos (stated below) is true for the group Zp ⊕ Zp, for all primes p > 4.67 × 1034.

Distribution of a Subset of Non-residues Modulo p

In this article, we prove that the sequence consisting of quadratic non-residues which are not primitive root modulo a prime p obeys Poisson law whenever \(\displaystyle \frac{p-1}{2}- \phi (p-1)\) is reasonably large as a function of p. To prove this, we count the number of \(\ell \)-tuples of quadratic non-residues which are not primitive roots mod p, thereby generalizing one of the results obtained in Gun et al. (Acta Arith, 129(4):325–333, 2007, [8]).

On the parity of the Fourier coefficients of j -function

Klein’s modular j-function is defined to be j(z) = E 4/Δ(z) = 1 q + 744 + ∞ ∑ n=1 c(n)q where z ∈ C with (z) > 0, q = exp(2iπz), E4(z) denotes the normalized Eisenstein series of weight 4 and Δ(z) is the Ramanujan’s Delta function. In this short note, we show that for each integer a ≥ 1, the interval (a, 4a(a+1)) (respectively, the interval (16a−1, (4a+1)2)) contains an integer n with n ≡ 7 (mod 8) such that c(n) is odd (respectively, c(n) is even).

The Length of an Arithmetic Progression Represented by a Binary Quadratic Form

Abstract In this paper we prove that if Q(x,y) = ax2 + bxy+cy2 is an integral binary quadratic form with a nonzero, nonsquare discriminant d and if Q represents an arithmetic progression {kn+ℓ : n = 0, 1,…, R-1}, where k and ℓ are positive integers, then there are absolute constants C1 >0 and L1 >0

ON SHORT ZERO-SUM SUBSEQUENCES II

such that R < C1ℓ (k2|d|)L1. Moreover, we prove that every nonzero integral binary quadratic form represents a nontrivial

On zero-sum sequences of prescribed length

3

Liouville numbers and Schanuel’s Conjecture

-term arithmetic progression infinitely often.