# Periodica Math. Hungar. -- to appear

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

### Diviseurs Des Nombres Ellipséphiques

Sylvain Col

Communicated by András Sárközy

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.

Received May 26, 2006; Accepted September 19, 2006
AMS Subject Classification (1991): ????

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

### The number of triangular islands on a triangular grid

Eszter K. Horváth , Zoltán Németh
 and
Gabriella Pluhár

Communicated by Mária B. Szendrei

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.

Received May 28, 2008; Accepted July 22, 2008
AMS Subject Classification (1991): 06D99, 05A05

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

### An application of index forms in cryptography

Attila Bérczes
 and
István Járási

Communicated by Attila Pethõ

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.

Received May 28, 2008; Accepted July 22, 2008
AMS Subject Classification (1991): 94A60, 11Y16

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

### The strip of minimum width covering a centrally symmetric set of points

Mario Huicochea
 and
Jesús Jerónimo-Castro2

Communicated by Imre Bárány

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 ${\cal P}.$

Received June 12, 2008; Accepted July 28, 2008
AMS Subject Classification (1991): ????
2 Supported by CONACYT, SNI 38848

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

### On the family of Diophantine triples $\delimiter "4266308 k+1,4k,9k+3\delimiter "5267309$

Bo He
 and
Alain Togbé

Communicated by Attila Pethõ

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 $d=144k^3 + 192k^2 + 76k+ 8$.

Received June 24, 2008; Accepted July 28, 2008
AMS Subject Classification (1991): 11D09, 11D25, 11J86

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

### Series and product representations for some mathematical constants

Horst Alzer
 and
Stamatis Koumandos

Communicated by Attila Pethõ

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 $S(m)=\sum _{k=1}^{\infty }{1\over 2^k+m}$.

Received July 2, 2008; Accepted October 31, 2008
AMS Subject Classification (1991): 11A67

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

### Combinatorial numbers in binary recurrences

Tünde Kovács

Communicated by Attila Pethõ

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.

Received July 24, 2008; Accepted October 31, 2008
AMS Subject Classification (1991): 11B37, 11B83, 11Y50

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

### Concatenation of pseudorandom binary sequences

Katalin Gyarmati1

Communicated by Attila Pethõ

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.

Received July 28, 2008; Accepted November 5, 2008
AMS Subject Classification (1991): 11K45
1 Research partially supported by Hungarian NFSR, Grants No. K49693, K67676, K72264 and the János Bolyai Research Fellowship.

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

### Large regular simplices contained in a hypercube

Hiroshi Maehara1 , Imre Z.\ Ruzsa2
 and
Norihide Tokushige3

Communicated by Imre Bárány

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 $n$.

Received September 4, 2008; Accepted December 2, 2008
AMS Subject Classification (1991): 52C07, 05B20
1 Supported by MEXT Grant-in-Aid for Scientific Research (B) 20340022.
2 Supported by Hungarian NFSR (OTKA), grants No. K 61908, K 72731.
3 Supported by MEXT Grant-in-Aid for Scientific Research (B) 20340022.