The Z-densities of the Fibonacci sequence.

摘要

Paul S. Bruckman and Peter G. Anderson made a conjecture about the Z-densities of the Fibonacci sequence, F(n), based on computational results. For a prime p, Z(p) is the "Fibonacci entry-point of n" or the smallest positive integer n such that p divides F(n), M(m,x) is the number of primes p is less than x such that m divides Z(p), and pi(x) is the number of primes less than x. We may define the "Z-density of m" to be Z(m) is the limit of x to infinity of M(m,x) divided by pi(x). The conjecture gives a formula for Z(m) for all positive m. We will prove the conjecture of Bruckman and Anderson.