Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence

Abstract

In mathematics, the prime counting function $\pi(x)$ is defined as the function yielding the number of primes less than or equal to a given number $x$. In this paper, we prove that the asymptotic limit of a summation operation performed on a unique subsequence of the prime numbers yields the prime number counting function $\pi(x)$ as $x$ approaches $\infty$. We also show that the prime number count $\pi(n)$ can be estimated with a notable degree of accuracy by performing the summation operation on the subsquence up to a limit $n$.