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Verification of the Firoozbakht conjecture for primes up to four quintillion

Alexei Kourbatov

Abstract

If $p_k$ is the k-th prime, the Firoozbakht conjecture states that the sequence $(p_k)^{1/k}$ is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify the Firoozbakht conjecture for primes up to four quintillion $(4\times10^{18})$.

Verification of the Firoozbakht conjecture for primes up to four quintillion

Abstract

If is the k-th prime, the Firoozbakht conjecture states that the sequence is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify the Firoozbakht conjecture for primes up to four quintillion .

Paper Structure

This paper contains 4 sections, 15 equations.

Table of Contents

  1. Introduction
  2. Computational verification for small primes
  3. What if we do not know $k=\pi(p_k)$?
  4. Verification for all gaps of a given size $g$