An explicit result for primes between cubes

Adrian W. Dudek*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    18 Citations (Scopus)

    Abstract

    We prove that there is a prime between n3 and (n + 1)3 for all n ≥ exp(exp(33:3)). This is done by first deriving the Riemann-von Mangoldt explicit formula for the Riemann zetafunction 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 ≥ 5×109. Notably, many of the explicit estimates developed in this paper can also find utility elsewhere in the theory of numbers.

    Original languageEnglish
    Pages (from-to)177-197
    Number of pages21
    JournalFunctiones et Approximatio, Commentarii Mathematici
    Volume55
    Issue number2
    DOIs
    Publication statusPublished - 2016

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