Explicit Estimate on Primes between Consecutive Cubes

TL;DR

This paper provides an explicit version of Ingham's Theorem to show that there is at least one prime between consecutive cubes for sufficiently large x, specifically when log log x is at least 15.

Contribution

It offers an explicit estimate confirming the existence of primes between consecutive cubes, improving understanding of prime distribution in specific intervals.

Findings

At least one prime exists between x^3 and (x+1)^3 for large x.

The condition log log x ≥ 15 ensures the prime's existence.

Provides explicit bounds related to prime distribution in cubic intervals.

Abstract

We give an explicit form of Ingham's Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes $x\sp{3}$ and $(x+1)\sp{3}$ if $\log\log x\ge 15$.