The m-th Cullen number Cm is a number of the form m2 m + 1 and the m-th Woodall number Wm has the form m2 m − 1. In 2003, Luca and Stănică proved that the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. A generalization of these sequences is defined by Cm,s = ms m+1 and Wm,s = ms m−1, for s > 1. In this pa...

This report is devoted to introduction in multichannel algorithm based on generalized numeration notations (GPN). The internal, external and mixed account are entered. The concept of the GPN and it’s classification as decomposition of an integer on composed of integers is discussed. Realization of multichannel algorithm on the basis of GPN is introduced. In particular, some properties of Fibona...

Data compression has been widely applied in many data processing areas. Compression methods use variable-size codes with the shorter codes assigned to symbols or groups of symbols that appear in the data frequently. Fibonacci coding, as a representative of these codes, is used for compressing small numbers. Time consumption of a decompression algorithm is not usually as important as the time of...

A number of keystream generators have been proposed which are based on Fibonacci sequences, and at least one has been elded. They are attractive in that they can use some of the security results from the theory of shift register based keystream generators, while running much more quickly in software. However, new designs bring new risks, and we show how a system proposed at last year's workshop...

The search for a local maximum of a function f(x) involves a sequence of function evaluations, i.e.s observations of the value of f(x) for a fixed value of x. A sequential search scheme allows us to evaluate the function at different points, one after the other, using information from earlier evaluations to decide where to locate the next ones. At each stage, the smallest interval in which a ma...

A Wieferich prime is a prime p such that 2p−1 ≡ 1 (mod p2). Despite several intensive searches, only two Wieferich primes are known: p = 1093 and p = 3511. This paper describes a new search algorithm for Wieferich primes using double-precision Montgomery arithmetic and a memoryless sieve, which runs significantly faster than previously published algorithms, allowing us to report that there are ...

a non-abelian finite group is called sequenceable if for some positive integer , is -generated ( ) and there exist integers such that every element of is a term of the -step generalized fibonacci sequence , , , . a remarkable application of this definition may be find on the study of random covers in the cryptography. the 2-step generalized sequences for the dihedral groups studied for their pe...

In this paper we introduce a family of infinite words that generalize the Fibonacci word and we study their combinatorial properties. We associate with this family of words a family of curves that are like the Fibonacci word fractal and reveal some fractal features. Finally, we describe an infinite family of polyominoes stems from the generalized Fibonacci words and we study some of their geome...