An Explicit Result for Primes Between Cubes

2014/01/17 by Adrian W. Dudek, Adrian Dudek, Dudek, Adrian · 3 citations
Mathematics · #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1401.4233

Abstract

We prove that there is a prime between n³ and (n+1)³ for all n ≥ exp(exp(33.217)). Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive mth powers for m ≥ 4.971 × 10⁹.

Citations

• The Difference Between Consecutive Primes, II
• Short effective intervals containing primes
• An explicit density estimate for Dirichlet 𝐿-series

Cited by

• Resolution of two conjectures by Erdős and Hall concerning separable numbers
• On the Maximal Gap between Primes
• Effective short intervals containing primes

Related

• Explicit Estimates in the Theory of Prime Numbers
• Explicit Estimate on Primes between Consecutive Cubes
• Cubic Waring-Goldbach problem with Piatetski-Shapiro primes
• Some explicit estimates for the error term in the prime number theorem
• Primes In Cubic Arithmetic Progressions