The dying rabbit problem revisited

In this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in the paper Hoggatt, V. E., Jr.; Lind, D. A. "The dying rabbit problem". Fibonacci Quart. 7 1969 no. 5, 482--487) and we calculate explicitly their general term (extending the work in the paper Miles, E. P., Jr. Generalized Fibonacci numbers and associated matrices. Amer. Math. Monthly 67 1960 745--752). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences.Comment: 8 pages, submitted to ANTS-VII

Origami constructions

A characterization of real numbers constructible by paper folding.Comment: 5 pages, 3 figure

A new look at the trailing zeroes on N!N!

Let us denote by $Z_b(n)$ the number of trailing zeroes in the base $b$ expansion 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 asymptotic aspects and we will give families of integers that do not belong to the image of $Z_b$

Counting domino trains

In this paper we present a way to count the number of trains that we can construct with a given set of domino pieces. As an application we obtain a new method to compute the total number of eulerian paths in an undirected graph as well as their starting and ending vertices

The ff-belos

The \emph{arbelos} is the shape bounded by three mutually tangent semicircles with collinear diameters. Recently, Sondow introduced the parabolic analog, the \emph{parbelos} and proved several properties of the parbelos similar to properties of the arbelos. In this paper we give one step further and generalize the situation considering the figure bounded by (quite) arbitrary similar curves, the \emph{$f$-belos}. We prove analog properties to those of the arbelos and parbelos and, moreover, we characterize the parbelos and the arbelos as the $f$-beloses satisfying certain conditions

On arithmetic numbers

An integer $n$ is said to be \textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, in Section 2, we give an interesting characterization of Mersenne numbers

A determinant of generalized Fibonacci numbers

We evaluate a determinant of generalized Fibonacci numbers, thus providing a common generalization of several determinant evaluation results that have previously appeared in the literature, all of them extending Cassini's identity for Fibonacci numbers

On the varieties of representations and characters of a family of one-relator subgroups. Their irreducible components

Let us consider the group $G =$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ 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 consequence, a formula for computing the number of irreducible components of $X(G)$ as a function of $m$ and $n$ is given. We provide a combinatorial description of $X(G)$ and we prove that in most cases it is possible to recover $(m,n)$ from the combinatorial structure of $X(G)$. Finally we compute the number of irreducible components of $R(G)$ and study the behavior of the projection $t:R(G)\longrightarrow X(G)$

On the last digit and the last non-zero digit of nnn^n in base bb

In this paper we study the sequences defined by the last and the last non-zero digits of $n^n$ in base $b$. For the sequence given by the last digits of $n^n$ in base $b$, we prove its periodicity using different techniques than those used by W. Sierpinski and R. Hampel. In the case of the sequence given by the last non-zero digits of $n^n$ in base $b$ (which had been studied only for $b=10$) we show the non-periodicity of the sequence when $b$ is an odd prime power and when it is even and square-free. We also show that if $b=2^{2^s}$ the sequence is periodic and conjecture that this is the only such case

A primality test for Kpn+1Kp^n+1 numbers

In this paper we generalize the classical Proth's theorem for integers of the form $N=Kp^n+1$. For these families, we present a primality test whose computational complexity is $\widetilde{O}(\log^2(N))$ and, what is more important, that requires only one modular exponentiation similar to that of Fermat's test. Consequently, the presented test improves the most often used one, derived from Pocklington's theorem, which usually requires the computation of several modular exponentiations together with some GCD's