On Fibonacci Quaternions

Fibonacci Sequences of Quaternions

In this paper Fibonacci sequences of quaternions are introduced, generalizing Fibonacci sequences over commutative rings, and properties of such sequences are investigated. In particular we are concerned with two kinds of Fibonacci sequences of generalized quaternions over finite fields Fp, with p an odd prime, and their periods.

ON QUATERNIONS By

On Fibonacci Powers

Fibonacci numbers have engaged the attention of mathematicians for several centuries, and whilst many of their properties are easy to establish by very simple methods, there are several unsolved problems connected to them. In this paper we review the history of the conjecture that the only perfect powers in Fibonacci sequence are 1, 8, and 144. Afterwards we consider more stronger conjecture an...

Notes on Fibonacci Partitions

Abstract. Let f1 = 1, f2 = 2, f3 = 3, f4 = 5, . . . be the sequence of Fibonacci numbers. It is well known that for any natural n there is a unique expression n = fi1 + fi2 + · · · + fiq , such that ia+1 − ia > 2 for a = 1, 2, . . . , q− 1 (Zeckendorf Theorem). By means of it we find an explicit formula for the quantity Fh(n) of partitions of n with h summands, all parts of them are the pairwis...

On Fibonacci Keystream Generators

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...