Volume 12, Issue 1

Abstract. The behavior of the multiplicative acceleration of Selmer's algorithm is widely unknown and no general result on convergence has been detected yet. Solely for its 2-dimensional, periodic expansions, there exist some results on convergence and approximation due to Fritz Schweiger. In this paper we show that periodic expansions of any dimension do in fact converge and that the coordinates of the limit points are rational functions of the largest eigenvalue of the periodicity matrix.

Abstract. In this article, we present two infinite dimensional matrices whose entries are recursively defined, and show that the sequence of their principal minors form the Lucas sequence, that is (2,1,3,4,7,...)$(2, 1, 3, 4, 7, \ldots )$. It is worth mentioning that to construct these matrices we use nonhomogeneous recurrence relations.

Abstract. We give a combinatorial proof of Guo's multi-generalization of Munarini's identity, answering a question of Guo.

Abstract. For a fixed prime p≡1(mod4)$p\equiv 1\hspace*{-1.42262pt}\pmod {4}$, we define the corresponding quartic residue graph and determine the number of triangles contained in such a graph. Our computation requires us to compute the number of pairs of consecutive quartic residues modulo p$p$ via the evaluation of certain quartic Jacobi sums.

Abstract. We generalize a result of Prodinger in a recent issue of Integers about the oscillatory behavior of a double summation related to the 2-adic valuation of the Boros–Moll sequence.

Abstract. We study a relation between factorials and their additive analog, the triangular numbers. We show that there is a positive integer k$k$ such that n!=2kT$n!=2^k T$ where T$T$ is a product of triangular numbers. We discuss the primality of T±1$T \pm 1$ and the primality of |T-p|$|T-p|$ where p$p$ is either the smallest prime greater than T$T$ or the greatest prime less than T$T$.

Abstract. We deduce new properties of the orbicyclic function E$E$ of several variables investigated in a recent paper by V. A. Liskovets. We point out that the function E$E$ and its connection to the number of solutions of certain linear congruences occur in the literature in a slightly different form. We investigate another similar function considered by Deitmar, Koyama and Kurokawa by studying analytic properties of some zeta functions of Igusa type. Simple number-theoretic proofs for some known properties are also given.

Abstract. For k≥1$k\ge 1$, let pk(n)$p_k(n)$ count the number of k$k$-component multipartitions of a nonnegative integer n$n$, and let σ(n)=∑d∣nd$\sigma (n) = \sum _{d\mid n}d$ be the usual divisor function. In this paper, we give a combinatorial proof of the recursive formula pk(n)=kn∑r=1npk(n-r)σ(r),$ p_k(n) = \frac{k}{n}\sum _{r=1}^n p_k(n-r)\sigma (r), $ both for k≥1${k\ge 1}$, where pk(n)$p_k(n)$ is defined as above, and also for k<0${k&lt;0}$, which requires a subtler approach. This formula was used by Gandhi in 1963 to prove several theorems, which yield numerous Ramanujan type congruences for pk(n)$p_k(n)$, including some well-known congruences for Ramanujan's τ$\tau $-function.

Abstract. Here we show that the only aliquot cycle consisting only of rep-digits in base 10 is the cycle consisting of the perfect number 6. Generally, we show that if g$g$ is an even positive integer, then there are only finitely many aliquot cycles consisting entirely of repdigits in base g$g$, which are, at least in principle, effectively computable.

Abstract. Let S(n)$S(n)$ denote the total number of digits `1' in the binary expansions of the integers between 1 and n-1${n-1}$. The Trollope–Delange formula is a classical result which provides an explicit representation for S(n)$S(n)$ in terms of the continuous, nowhere differentiable Takagi function. Recently, connections have been established between digital sums such as S(n)$S(n)$ and certain functional equations associated with the Takagi function and its relatives. In the present paper we explore such a connection to derive a new, simple proof for the Trollope–Delange formula as well as for some of its generalizations involving power and exponential sums.