Theoretical Mathematics & Applications

Improved version of: From Sierpinskis conjecture to Legendre’s

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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 Sierpinski’s conjecture of section 3.5. It includes a full rewriting of sections 3.3 to 3.5 as well as other minor changes. Legendre’s 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 Oppermann’s conjecture (1882), Sierpi´nski’s S conjecture (1958) as well as Legendre’s. Finally, as consequences, p_m+1 -p_m = O(SR(p_m)), Andrica’s (1986) and Brocard’s (1904) conjectures are proved.