Fibonacci and Lucas numbers have lo

Fibonacci and Lucas numbers have long interested mathematicians for their intrinsic theory and their applications. For
rich applications of these numbers in science and nature, one can see the citations in [15]. For instance, the ratio of two
consecutive of these numbers converges to the Golden section  D 1C
p
5
2 . (The applications of Golden ratio appears in many
research areas, particularly in Physics, Engineering, Architecture, Nature and Art. Physicists Naschie and Marek-Crnjac gave
some examples of the Golden ratio in Theoretical Physics and Physics of High Energy Particles [69]). Therefore, in this paper,
we are mainly interested in whether some new mathematical developments can be applied to these numbers. In this paper
we obtain new results about Lucas numbers. As a reminder for the rest of this paper, for n > 2, the well-known Fibonacci
fFng and Lucas fLng sequences are defined by Fn D Fn