Theoretical Mathematics & Applications

Improved version of: From Sierpinskis conjecture to Legendreís

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  • Abstract

     

    This improved version of the original article published in vol.4, no.2, 2014, 65-77 of this journal is published here in full to close a gap in the proof of Sierpinskiís conjecture of section 3.5. It includes a full rewriting of sections 3.3 to 3.5 as well as other minor changes. Legendreís conjecture (supposedly 1808) states that: There is always at least one prime number between two consecutive squares N2 and (N + 1)2 for any integer N > 0. In the present article, an elementary proof of this conjecture is given by creating and solving a D conjecture that includes Oppermannís conjecture (1882), Sierpi´nskiís S conjecture (1958) as well as Legendreís. Finally, as consequences, pm+1 -pm = O(SR(pm)), Andricaís (1986) and Brocardís (1904) conjectures are proved.