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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 us denote by Z b (n) the number of trailing zeroes in the base b expansión of n!. In this paper we study with some detail the behavior of the function Z b. In particular, since Z b is non-decreasing, we will characterize the points where it increases and we will compute the amplitude of the jump in each of such points. In passing, we will study some… (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)

- Jorge Mart́ın-Morales, Antonio M. Oller-Marcén
- 2008

Let us consider the group G = x, y | x m = y n with m and n nonzero integers. In this paper, we study the variety of representations R(G) and the character variety X(G) in SL(2, C) of the group G, obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to X(G) in the coordinates X = t x , Y = t y and Z = t xy. As an easy… (More)

- Jorge Mart́ın-Morales, Antonio M. Oller-Marcén
- 2008

Let us consider the group G = x, y | x m = y n with m and n nonzero integers. In this paper, we study the variety of representations R(G) and the character variety X(G) in SL(2, C) of the group G, obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to X(G) in the coordinates X = t x , Y = t y and Z = t xy. As an easy… (More)

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