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An Explicit Result for Primes Between Cubes
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abstract
We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. 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 $m$th powers for $m \geq 4.971 \times 10^9$.
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Cited by 1 Pith paper
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Effective short intervals containing primes
For every n ≥ 106 and every x ≥ 1, there is always a prime in the interval [x, x + x^(1−1/n)], making several classical prime-gap results fully explicit.
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