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Almost prime numbers

Tigran Hakobyan

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 $ω(p-1)\leq 2$.

Almost prime numbers

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 .
Paper Structure (13 sections, 17 theorems, 36 equations)

This paper contains 13 sections, 17 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Proof of Theorem 1.4
  4. Proof of Theorem 1.5
  5. Proof of Theorem 1.7 and corollaries
  6. Proof of Theorem 1.7
  7. Proof of Corollary 1.8
  8. Proof of Corollary 1.10
  9. Proof of Corollary 1.11
  10. Proof of Theorem 1.12 and corollaries
  11. Proof of Theorem 1.12
  12. Proof of Corollary 1.13
  13. Proof of Corollary 1.14

Key Result

Theorem 1.4

Let $n$ be an almost-prime number. Then

Theorems & Definitions (41)

  • Conjecture
  • Definition 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • ...and 31 more