For a distribution F * τ of a random sum S τ = ξ 1 +. .. + ξ τ of i.i.d. random variables with a common distribution F on the half-line [0, ∞), we study the limits of the ratios of tails F * τ (x)/F (x) as x → ∞ (here τ is an independent counting random variable). We also consider applications of obtained results to random walks, compound Poisson distributions, infinitely divisible laws, and su...

In this paper, the divisibility property of type 2 (p; q)-analogue r-Whitney numbers second kind is established. More precisely, a congruence relation modulo pq for derived.

Irmak recently asked an open question related to divisibility properties of Fibonacci and Lucas quaternions [4, p. 374]. In this paper, we give answer hybrid number version question.

This note presents some results on the divisibility of certain coe"cients of the chromatic polynomials. c © 2003 Elsevier B.V. All rights reserved.

1 Integer arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Modular arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 The RSA cryptosystem . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 Primality testing . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

For given integers a, b and j ≥ 1 we determine the set R (j) a,b of integers n for which a n − b is divisible by n . For j = 1, 2, this set is usually infinite; we determine explicitly the exceptional cases for which a, b the set R (j) a,b (j = 1, 2) is finite. For j = 2, we use Zsigmondy’s Theorem for this. For j ≥ 3 and gcd(a, b) = 1, R (j) a,b is probably always finite; this seems difficult ...

In this chapter we will be investigating the intricacies of a seemingly innocuous mathematical object Pascal’s Triangle. After a brief refresher of our understanding of the triangle, we will delve into Cellular Automata as a method of building patterns, and then relate this to a specific pattern within Pascal’s Triangle. Then we shall investigate specific divisibility identities that can aid in...