We characterize Giuga numbers as solutions to the equation n ′ = an + 1, with a ∈ N and n ′ being the arithmetic derivative. Although this fact does not refute Lava's conjecture, it does suggest doubts about its veracity.
Generalized Cullen Numbers are positive integers of the form C b (n) := nb n + 1. In this work we generalize some known divisibility properties of Cullen Numbers and present two primality tests for this family of integers. The first test is based in the following property of primes from this family: n b n ≡ (−1) b (mod nb n + 1). It is stronger and has less… (More)
Let Z b (n) denote the number of trailing zeroes in the base-b expansion of n!. In this paper we study the connection between the expression of ϑ(b) := lim n→∞ Z b (n)/n in base b, and that of Z b (b k). In particular, if b is a prime power, we will show the equality between the k digits of Z b (b k) and the first k digits in the fractional part of ϑ(b). In… (More)