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      One-Click Submission System Now Available for SO Preprints, learn more on how this works in our blog post and don't forget to check the video, too!

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      Version and Review History
      Preprint version 2
      Preprint version 1
       
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      Note for the Prime Gaps

      Preprint
      In review
      research-article
      Author(s):   1 ,
      Publication date (Electronic preprint):
      Journal: ScienceOpen Preprints
      Publisher: ScienceOpen
      Keywords: prime gaps, prime numbers, Chebyshev function, primorial numbers
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            Revision notes

            I think Lemma 3 is flawed in the previous version. However, I was able to overcome that problem and demonstrate a partial result. In fact, in this new version we show that the Andrica conjecture should be true for every sufficiently large gap between primes. We changed the abstract, keywords, content and references. This current version remains a good contribution to the study of the distribution of prime numbers in their gaps.

            Abstract

            Content

            Author and article information

            Journal
            Title: ScienceOpen Preprints
            Publisher: ScienceOpen
            Publication date (Electronic preprint): 17 May 2024
            Affiliations
            [1 ] GROUPS PLUS TOURS INC., 9611 Fontainebleau Blvd, Miami, FL, 33172, USA;
            Author notes
            Author information
            https://orcid.org/0000-0001-8210-4126
            Article
            DOI: 10.14293/PR2199.000837.v2
            SO-VID: 604a88b4-0722-47b5-87d0-6faf11d5355c
            License:

            This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

            History
            Date received : 30 April 2024
            Categories

            Data availability: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
            ScienceOpen disciplines: General mathematics,Number theory
            Keywords: prime gaps,prime numbers,Chebyshev function,primorial numbers

            References

            1. Jackson Terence. 95.42 Irrational square roots of natural numbers — a geometrical approach. The Mathematical Gazette. Vol. 95(533):327–330. 2011. Cambridge University Press (CUP). [Cross Ref]

            2. Rosser J. Barkley, Schoenfeld Lowell. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics. Vol. 6(1)1962. Duke University Press. [Cross Ref]

            3. Platt D. J., Trudgian T. S.. On the first sign change of $\theta (x) -x$. Mathematics of Computation. Vol. 85(299):1539–1547. 2015. American Mathematical Society (AMS). [Cross Ref]

            4. Nagura Jitsuro. On the interval containing at least one prime number. Proceedings of the Japan Academy, Series A, Mathematical Sciences. Vol. 28(4)1952. Project Euclid. [Cross Ref]

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