Periodica Math. Hungar. -- to appear

Periodica. Math.
58(2009), 1--23

Abstract. Les propriétés multiplicatives des nombres ellipséphiques peuvent {\accent 94 e}tre obtenues {\accent 18 a} l'aide des moments de la série génératrice de cette suite. Nous donnons des estimations précises pour les grands moments par deux méthodes distinctes : l'une combinatoire fournit un résultat précis dans le cas réputé le plus difficile des nombres n'utilisant que les $0$ et les $1$ ; la seconde purement analytique fournit un résultat sans condition sur les chiffres.

Periodica. Math.
58(2009), 25--34

Abstract. The aim of the present paper is to carry on the research of Czédli in determining the maximum number of rectangular islands on a rectangular grid. We estimate the maximum of the number of triangular islands on a triangular grid.

Periodica. Math.
58(2009), 35--45

Abstract. We investigate the possibility of using index forms as basic ingredients of cryptographically important functions. We suggest the use of a hash function based on index forms and we prove some important properties of the suggested function.

Periodica. Math.
58(2009), 47--58

Abstract. In this paper the following is proved: let ${\cal P}$ be a centrally symmetric set of points, such that the distance between any pair of points is at least $1$ and every three of them can be covered by a strip of width $1$. Then there is a strip of width $\radical "270370 {2}$ covering ${\cal P}.$

Periodica. Math.
58(2009), 59--70

Abstract. We prove that if $k$ is a positive integer and $d$ is a positive integer such that the product of any two distinct elements of the set $ \delimiter "4266308 k+1,4k,9k+3,d\delimiter "5267309 $ increased by $1$ is a perfect square, then $d=144k^3 + 192k^2 + 76k+ 8$.

Periodica. Math.
58(2009), 71--82

Abstract. We present several series and product representations for $\gamma $, $\pi $, and other mathematical constants. One of our results states that we have for all real numbers $\mu >0$: $$ \gamma = \sum _{k=0}^{\infty }{1\over (1+\mu )^{k+1}} \sum _{m=0}^k{k \atopwithdelims ()m}(-1)^m {\mu }^{k-m}S(m),$$ where $S(m)=\sum _{k=1}^{\infty }{1\over 2^k+m}$.

Periodica. Math.
58(2009), 83--98

Abstract. We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.

Periodica. Math.
58(2009), 99--119

Abstract. In the applications it may occur that our initial pseudorandom binary sequence turns out to be not long enough, thus we have to take the concatenation or merging of it with another pseudorandom binary sequences. Here our goal is study when can we form the concatenation of several pseudorandom binary sequences belonging to a given family? We introduce and study new measures which can be used for answering this question.

Periodica. Math.
58(2009), 121--126

Abstract. We prove that the $n$-dimensional unit hypercube contains an $n$-dimensional regular simplex of edge length $c\radical "270370 {n}$, where $c>0$ is a constant independent of $n$.