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
. (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
and Lucas sequences are defined by Fn D Fn