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Application of the Inclusion-Exclusion Principle to Prime Number Subsequences

Michael P. May

Abstract

We apply the Inclusion-Exclusion Principle to a unique pair of prime number subsequences to determine whether these subsequences form a small set or a large set and thus whether the infinite sum of the inverse of their terms converges or diverges. In this paper, we analyze the complementary prime number subsequences P(prime) and P(double-prime) as well as revisit the twin prime subsequence P2.

Application of the Inclusion-Exclusion Principle to Prime Number Subsequences

Abstract

We apply the Inclusion-Exclusion Principle to a unique pair of prime number subsequences to determine whether these subsequences form a small set or a large set and thus whether the infinite sum of the inverse of their terms converges or diverges. In this paper, we analyze the complementary prime number subsequences P(prime) and P(double-prime) as well as revisit the twin prime subsequence P2.
Paper Structure (9 sections, 2 theorems, 73 equations, 3 tables)

This paper contains 9 sections, 2 theorems, 73 equations, 3 tables.

Table of Contents

  1. The Prime Subsequences ${\mathbb{P{'}}}$ and ${\mathbb{P{"}}}$
  2. Asymptotic Densities of ${\mathbb{P{'}}}$ and ${\mathbb{P{"}}}$
  3. ${\pi{'(x)}}$ and ${\pi{"(x)}}$
  4. ${\pi{(x)}}$ via the Inclusion-Exclusion Principle
  5. Estimating ${\pi{"(x)}}$
  6. Estimating ${\pi{'(x)}}$
  7. Estimating ${\pi{_2(x)}}$
  8. Mathematica calculations
  9. Conclusion

Key Result

Theorem 1

If $x \geq 2$, then ${\prod_{p\leq x} \left( 1 - \dfrac{1}{p} \right)} < \dfrac{1}{\ln{x}}$.

Theorems & Definitions (2)

  • Theorem 1
  • Lemma 1