We extend results of [1, 2, 3] on the visibility problem for lattice points in Zd to the case of function fields over finite fields which are related to important questions regarding the corresponding q-Jacobsthal function. Résumé. Nous étendons résultats de [1, 2, 3] sur le probleme de la visibilité des points du réseau entier Zd au cas des corps de fonctions sur un corps fini, en rapport avec...

For any integer n ≥ 1, let j(n) denote the Jacobsthal function, and ω(n) the number of distinct prime divisors of n. In 1962 Jacobsthal conjectured that for any integer r ≥ 1, the maximal value of j(n) when n varies over N with ω(n) = r is attained when n is the product of the first r primes. We show that this is true for r ≤ 23 and fails at r = 24, thus disproving Jacobsthal’s conjecture.

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the i, j entry of A A is adjacency matrix is equal to the number of walks of length m from vertex i to vertex j, we s...

In this paper we consider the generalized Jacobsthal Jn,m and Jacobsthal-Lucas numbers jn,m. Also, introduce new sequences of An,m, Bn,m, Cn,m Dn,m. Namely, these are convolutions Further, find generating functions some recurrence relations for numbers.

It is well known that if p is a prime such that p ≡ 1 (mod 4), then p can be expressed as a sum of two squares. Several proofs of this fact are known and one of them, due to E. Jacobsthal, involves the identity p = x2 + y2, with x and y expressed explicitly in terms of sums involving the Legendre symbol. These sums are now known as the Jacobsthal sums. In this short note, we prove that if p ≡ 1...

We study several classes of polynomials, which are related to the Chebyshev, Morgan-Voyce, Horadam and Jacobsthal polynomials. Thus, we unify some of well-known results.