We formalized some basic properties of the Fibonacci numbers using definitions and lemmas from [7] and [23], e.g. Cassini’s and Catalan’s identities. We also showed the connections between Fibonacci numbers and Pythagorean triples as defined in [31]. The main result of this article is a proof of Carmichael’s Theorem on prime divisors of prime-generated Fibonacci numbers. According to it, if we ...

In this paper, we consider the generalized Fibonacci p-numbers and then we give the generalized Binet formula, sums, combinatorial representations and generating function of the generalized Fibonacci p-numbers. Also, using matrix methods, we derive an explicit formula for the sums of the generalized Fibonacci p-numbers. c © 2007 Elsevier Ltd. All rights reserved.

Among numerical sequences, the Fibonacci numbers have achieved a kind of celebrity status with such fabulous properties, it is no wonder that the Fibonacci numbers stand out as a kind of super sequence. Fibonacci numbers have been studied in many different forms for centuries and the literature on the subject is consequently, incredibly vast. The Fibonacci sequence has been generalized in a num...

In this paper we consider infinite sums derived from the reciprocals of the Fibonacci numbers, and infinite sums derived from the reciprocals of the square of the Fibonacci numbers. Applying the floor function to the reciprocals of these sums, we obtain equalities that involve the Fibonacci numbers.

In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Pell Narayana simultaneously. We prove some identities generalize well-known relations for numbers their generalizations. A matrix representation generalized is given, too.