In 2009, Luca and Nicolae [[Formula: see text], Integers 9 (2009) A30] proved that the only Fibonacci numbers whose Euler totient function is another number are [Formula: text]. 2015, Faye [Pell a Pell number, Publ. Inst. Math. 101(115) (2017) 231–245] text] Here, we add to these two results prove for any fixed natural if define sequence as all then solution Diophantine equation

We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let {bn}n=0 be the sequence of Fibonacci numbers. The bn-point Fibonacci set Fn ⊂ [0, 1] is defined as Fn := {(μ/bn, {μbn−1/bn})} μ=1, where {x} is the fractional part of a number x ∈ R. It is known that cubature formulas based on Fibonacci set Fn give optimal rate of error...

Zeckendorf’s theorem states that every positive integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers. This theorem induces a binary numeration system for the positive integers known as Fibonacci coding. Fibonacci code is a variable-length prefix code that is robust against insertion and deletion errors and is useful in data transmission and data compression. In this pape...

The Fibonacci dimension fdim(G) of a graph G is introduced as the smallest integer f such that G admits an isometric embedding into Γf , the f -dimensional Fibonacci cube. We give bounds on the Fibonacci dimension of a graph in terms of the isometric and lattice dimension, provide a combinatorial characterization of the Fibonacci dimension using properties of an associated graph, and establish ...

In a recent paper, Kalman [3] derives many interesting properties of generalized Fibonacci numbers. In this paper, we take a different approach and derive some other interesting properties of matrices of generalized Fibonacci numbers. As an application of such properties, we construct an efficient algorithm for computing matrices of generalized Fibonacci numbers. The topic of generalized Fibona...