Relationship Between Lucas And The Next Number In The Subject Of Prime Factors

This paper mainly introduces the historical background and the present research situation of the Fibonacci sequence and the Lucas sequence development. Simple instructions for solving the problem was given in this paper, then divisibility of the Lucas series was discussed, and then gained the relationship between the Lucas number containing factor2,3,4,6,9,11and subscript in the Lucas series.According to the definition of Lucas sequence, have discussed the relationship between the prime factorization2or3and its subscript by the elementary number theory and mathematical induction method, then obtained the following conclusions:The index factor2in the standard factorization of L3m depends on its subscript. when its subscript is even, The index factor2in the standard factorization index of L3m was1;when its subscript is odd, The index factor2in the standard factorization index of L3m was2. The factor3’s index in the standard factorization of Ln depends on the factor3’s index in the standard factorization of its subscript "n".In the end have gained the conclusion that the Lucas number’s sum of square is the Lucas numbers or Fibonacci numbers with using the Yang Hui triangle’s row as coefficient.