A perfect number is any positive integer that is equal to the sum of its proper divisors. Several years ago, F. Luca showed that the Fibonacci and Lucas numbers contain no perfect numbers. In this paper, we alter the argument given by Luca for the nonexistence of both odd perfect Fibonacci and Lucas numbers, by making use of an 1888 result of C. Servais. We also provide a brief historical accou...

Many combinatorialists live by Mach’s words, and take it as a personal challenge. For example, nearly all of the Fibonacci identities in [5] and [6] have been explained by counting arguments [1, 2, 3]. Among the holdouts are those involving infinite sums and irrational quantities. However, by adopting a probabilistic viewpoint, many of the remaining identities can be explained combinatorially. ...

This paper will explore the relationship between the Fibonacci numbers and the Euclidean Algorithm in addition to generating a generalization of the Fibonacci Numbers. It will also look at the ratio of adjacent Fibonacci numbers and adjacent generalized Fibonacci numbers. Finally it will explore some fun applications and properties of the Fibonacci numbers.

Secure message transmission is generally required for the system where transmitted message need to be verified at the receiver end. Fibonacci develops the reversible encryption algorithms as mentioned in below technique. The technique considers a message as binary string on which the Fibonacci Based Position Substitution (FBPS) method is applied. A block of n bits is taken as an input stream fr...

We study the discretised segments generated by the iterated Tribonacci substitution and the projections of the integer points on them to some plane. After suitable transformations we get a sequence of finite two-dimensional words which tends to a doubly rotational word on Z. (Without scaling we would get the Rauzy fractal.) As an introduction we start with the corresponding case of the Fibonacc...

The representations −→ N 1 + −→ N 2 = D of a natural number D as the sum of two Fibonacci-even numbers −→ N i = F1 ◦ Ni, where ◦ is the circular Fibonacci multiplication, are considered. For the number s(D) of solutions, the asymptotic formula s(D) = c(D)D + r(D) is proved; here c(D) is a continuous, piecewise linear function and the remainder r(D) satisfies the inequality |r(D)| ≤ 5 + ( 1 ln(1...

Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79–95, 2011) proved that at every position of an infinite Sturmian word starts an abelian power of exponent k, for every positive integer k. Here, we improve on this result, studying the maximal exponent of abelian powers and abelian repetitions (an abelian repetition is the analogous of a fractional power in the abelian setting) occurring in...

This paper describes how a subclass of the rational knots* may be constructed sequentially., the knots in the sequence having 19 29 ..., i s ... crossings. For these knots, the values of a certain knot invariant are Fibonacci numbers, the i knot in the sequence having invariant number Fi . The knot invariant has a wide number of interpretations and properties, and some of these will be outlined...

1. A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." The Fibonacci Quarterly 3 (1965):161-75. 2. A. F. Horadam. "Complex Fibonacci Numbers and Fibonacci Quaternions." Amer. Math. Monthly 70 (1963):289-91. 3. A. L. Iakin. "Generalized Quaternions with Quaternion Components." The Fibonacci Quarterly 15 (1977):35Q-52. 4. A. L. Iakin. "Generalized Quaternions of Higher...

The objective of this study was to develop a new optimal parallel algorithm for matrix multiplication which could run on a Fibonacci Hypercube structure. Most of the popular algorithms for parallel matrix multiplication can not run on Fibonacci Hypercube structure, therefore giving a method that can be run on all structures especially Fibonacci Hypercube structure is necessary for parallel matr...