Almost prime numbers

TL;DR

This paper introduces the concept of almost prime numbers, explores their properties, and conjectures their infinitude, potentially shedding light on the distribution of primes with limited prime factors in p-1.

Contribution

It generalizes prime numbers to almost primes, establishes properties, and proposes the conjecture of their infinitude, linking to prime distribution questions.

Findings

Composite almost primes, if they exist, are Carmichael numbers.

Several properties of almost primes are proven.

Conjecture that infinitely many composite almost primes exist.

Abstract

We introduce the concept of an almost prime number generalizing a prime number. It turns out that a composite almost prime number must be a Carmichael number, in case it exists. We prove several properties of almost prime numbers and conjecture the infinitude of composite almost primes. In this regards, the latter seems to hold the potential of shading light onto the infinitude of primes with $\omega(p-1)\leq 2$.