On a Bernoulli numbers conjectures.
On a binary recurrent sequence of polynomials
In this paper, we study the properties of the sequence of polynomials given by , for , where is non-constant and the characteristic of is . This complements some results from R. Euler, L.H. Gallardo: On explicit formulae and linear recurrent sequences, Acta Math. Univ. Comenianae, 80 (2011) 213-219.
On a certain class of arithmetic functions
A homothetic arithmetic function of ratio is a function such that for every . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of in terms of the period and the ratio of .
On a class of arithmetical sets
On a class of combinatorial Diophantine equations.
On a class of combinatorial sums involving generalized factorials.
On a class of densities of sets of positive integers.
On a class of polynomials with integer coefficients.
On a class of ternary inclusion-exclusion polynomials.
On a class of Thue-Morse type sequences.
On a combinatorial problem of Asmus Schmidt.
On a congruence of Emma Lehmer related to Euler numbers
A congruence of Emma Lehmer (1938) for Euler numbers modulo p in terms of a certain sum of reciprocals of squares of integers was recently extended to prime power moduli by T. Cai et al. We generalize this further to arbitrary composite moduli n and characterize those n for which the sum in question vanishes modulo n (or modulo n/3 when 3|n). Primes for which play an important role, and we present some numerical results.
On a conjecture concerning sequences of the third order
On a conjecture of Erdös and Heilbronn
On a conjecture of Hamidoune for subsequence sums.
On a conjecture of Lemke and Kleitman
On a conjecture of Morgan Ward, I
On a conjecture of Morgan Ward, II
On a conjecture of Morgan Ward, III
On a conjecture of Sárközy and Szemerédi
Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. In 1994, Sárközy and Szemerédi conjectured that there exist infinite additive complements A and B with lim sup A(x)B(x)/x ≤ 1 and A(x)B(x)-x = O(minA(x),B(x)), where A(x) and B(x) are the counting functions of A and B, respectively. We prove that, for infinite additive complements A and B, if lim sup A(x)B(x)/x ≤ 1, then, for any given M > 1,...