Fibonacci Coefficients And Small Factor Problems With Natural Numbers

In this thesis,we give the proofs of some identities on the sums of Fibonacci coefficients and study the problems that when the small divisors of a natural number are in geometrical progressions.1.Some identities on the sums of Fibonacci coefficients.In this paper,the Fibonacci coefficients are the generalized form of binomial coefficients,replacing the natural number n by the terms of a Fibonacci sequence {Fn},signed as[km]F=(FmFm-1…Fm-k+1)/(F1…Fk).Mathematicians have obtained many divisible properties and identities for the Fibonacci coefficients.Now,we prove some new recursive relations and identities on the sums of Fibonacci coefficients:(?)2.When the small divisors of a natural number are in geometrical progressions.For a natural number n,let Tn denote the set of positive divisors of n that is not greater than its square root,that is,Tn={d:d|n,d?(?)?.We say that Tn is the set of small divisors of n.We prove that three cases are true when Tn is equal to {aj}j=0k-1.Here are the three cases.(1)n=1;(2)n=p?,for some prime p,??1;(3)n=p?q,q>p?,??1.Furthermore,we come to a conclusion that the small divisors of n are not in geometrical progression if n=p?q?,?,?>1 or ?(n)? 3,where ?(n)means the number of different prime divisors of n.