Comparing the growth of the prime numbers to that of the natural numbers

Nathan Wakefield, CU Boulder
Abstract: Consider a function f(n):N->N which measures the "growth" of the natural numbers in the following sense: for the sequence {1,2,3,....}, we define f(n) by f(n):=max{k in N: (n!)^2 \geq(n+k)!}. This growth function has interesting asymptotic properties; we prove that when f is defined on the sequence of positive integers, {1,2,3,...}, we have lim\limits_{n-> \infty}\frac{f(n)}{n}=1. This idea can be generalized to any increasing sequence of integers. Specifically, we study this function defined on the sequence of prime numbers by considering the factorial like product of the sequence, and study the value of the associated limit.