In the paper sufficient conditions for the -density of a set of positive integers in terms of logarithmic densities are given. They differ substantially from those derived previously in terms of asymptotic densities.
We study the logarithmic frequency of letters and words in morphic sequences and show that this frequency must always exist, answering a question of Allouche and Shallit.
We study representation functions of asymptotic additive bases and more general subsets of ℕ (sets with few nonrepresentable numbers). We prove that if ℕ∖(A+A) has sufficiently small upper density (as in the case of asymptotic bases) then there are infinitely many numbers with more than five representations in A+A, counting order.
A positive is called a balancing number if We prove that there is no balancing number which is a term of the Lucas sequence.
We show that the only Lucas numbers which are factoriangular are and .
Let be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are and , respectively. We show that the Diophantine equation has only finitely many solutions in , where , is even and . Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral points on...