In this paper, using an argument of P. Erdős, K. Alniaçik and E. Saias, we extend earlier results on Liouville numbers, due to P. Erdős,

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Abstract. For integers m ≥ 2 and d ≥ 1, we study the set S m of m consecutive integers which satisfies the property that for each x ∈ S m there exists y ∈ S m such that gcd(x, y) > d. This problem was first posed and studied by S. S. Pillai for the case d = 1. In this article, we elaborate on an argument of T. Vijayaraghavan for d = 1 and of Y.For an integer m ≥ 2, let S m be a set of m consecutive integers. Let d ≥ 1 be an integer. We say that the set S m has property P d if there exists an element x ∈ S m such that gcd(x, y) ≤ d for all y ∈ S m with y = x. In this case, we also say that the element x has property P d . When no such element exists, we say that S m does not have property P d . Thus, if d = 1, S m has property P 1 means that there exists x ∈ S m which is co-prime to all other elements in S m .In 1940, S. S. Pillai (in [12]) and, independently, Szekeres (see for instance, [9]) first studied the problem of finding sets S m having property P 1 . Pillai was motivated by this problem while trying to solve a folklore conjecture that a product of two or more consecutive integers is never a perfect power.This remarkable result was proved by P. Erdős and J. L. Selfridge (in [6]) in 1975. Pillai (in [12]) showed that S m has property P 1 for m < 17. Thus, any set of consecutive integers having less than 17 elements has property P 1 . Further, Pillai succeeded in proving that for 17 ≤ m ≤ 430, there exist infinitely many sets S m for which property P 1 does not hold. To prove this result, he used sieving techniques and introduced numbers known as gap numbers.

For any integer nX3; by gðZ n "Z n Þ we denote the smallest positive integer t such that every subset of cardinality t of the group Z n "Z n contains a subset of cardinality n whose sum is zero. Kemnitz (Extremalprobleme fu¨r Gitterpunkte, Ph.D. Thesis, Technische Universita¨t Braunschweig, 1982) proved that gðZ p "Z p Þ ¼ 2p À 1 for p ¼ 3; 5; 7: In this paper, as our main result, we prove that gðZ p "Z p Þ ¼ 2p À 1 for all primes pX67: r 2004 Elsevier Inc. All rights reserved. MSC: primary 11B75; secondary 20K99

Let G be a finite abelian group. By OlðGÞ; we mean the smallest integer t such that every subset ACG of cardinality t contains a non-empty subset whose sum is zero. In this article, we shall prove that for all primes p44:67 Â 10 34 ; we have OlðZ p "Z p Þ ¼ p þ OlðZ p Þ À 1 and hence we have OlðZ p "Z p Þpp À 1 þ J ffiffiffiffiffi 2p p þ 5 log pn: This, in particular, proves that a conjecture of Erdo +s (stated below) is true for the group Z p "Z p for all primes p44:67 Â 10 34 : r

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, . . . , n} such that elements of A are incongruent modulo p and non-zero modulo p.

Let p be any odd prime number. Let k be any positive integer such that 2 ≤ k ≤ p+1 3 + 1. Let S = (a 1 , a 2 , . . . , a 2p−k ) be any sequence in Z p such that there is no subsequence of length p of S whose sum is zero in Z p . Then we prove that we can arrange the sequence S as follows:This extends a result in [13] to all primes p and k satisfying (p + 1)/4 + 3 ≤ k ≤ (p + 1)/3 + 1. Also, we prove that if g denotes the number of distinct residue classes modulo p appearing in the sequence S in Z p of length 2p − k (2 ≤ k ≤ [(p + 1)/4] + 1), and g ≥ 2 √ 2 √ k − 2, then there exists a subsequence of S of length p whose sum is zero in Z p .

Abstract. Let p be a prime number such that p ≡ 1 (mod r) for some integer r > 1. Let g > 1 be an integer such that g has order r in (Z/pZ) * . Letbe the g-adic expansion. Our result implies that the "average" digit in the g-adic expansion of 1/p is (g − 1)/2 with error term involving the generalized Bernoulli numbers B 1,χ (where χ is a character modulo p of order r with χ(−1) = −1). Also, we study, using Bernoulli polynomials and Dirichlet Lfunctions, the set equidistribution modulo 1 of the elements of the subgroup H n of (Z/nZ) * as n → ∞ whenever |H n |/ √ n → ∞.