Fibonacci integers

A Fibonacci integer is an integer in the multiplicative group generated by the Fibonacci numbers. For example, 77=21⋅55/(3⋅5) is a Fibonacci integer. Using some results about the structure of this multiplicative group, we determine a near-asymptotic formula for the counting function of the Fibonacci integers, showing that up to x the number of them is between exp(c(logx)1/2−(logx)ϵ) and exp(c(logx)1/2+(logx)1/6+ϵ), for an explicitly determined constant c. The proof is based on both combinatorial and analytic arguments.