New Formulas for Semi-Primes. Testing, Counting and Identification of the $n^{th}$ and next Semi-Primes

TL;DR

This paper introduces a new primality test and formulas for counting, identifying, and finding the next semi-prime, leveraging prime knowledge up to the cube root of N, advancing semi-prime analysis methods.

Contribution

It presents novel formulas for semi-prime counting and identification, along with a new semi-primality test, based on primes up to the cube root of N.

Findings

New semi-primality test developed

Formulas for counting semi-primes introduced

Methods for identifying and finding next semi-primes provided

Abstract

In this paper we give a new semiprimality test and we construct a new formula for $\pi ^{(2)}(N)$, the function that counts the number of semiprimes not exceeding a given number $N$. We also present new formulas to identify the $n^{th}$ semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots of $N : P_{1}, \; P_{2}....P_{\pi \left( \sqrt[3]{N}\right) }\leq \sqrt[3]{N}$.