Number Theory


A Conjecture of Existence of Prime Numbers in Arithmetic Progressions

Authors: Juan Moreno Borrallo

In this paper it is proposed and proved a conjecture of existence of a prime number on the arithmetic progression S_{a,b}=\left\{ ab+1,ab+2,ab+3,...,ab+(b-1)\right\} As corollaries of this proof, they are proved many classical prime number’s conjectures and theorems, but mainly Bertrand's theorem, and Oppermann's, Legendre’s, Brocard’s, and Andrica’s conjectures. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated a corollary which implies some advance on the conjecture of the existence of infinite prime numbers of the form n^{2}+1.

Comments: 18 Pages.

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Submission history

[v1] 2018-01-16 14:33:02 (removed)
[v2] 2018-01-20 09:13:54 (removed)
[v3] 2018-06-26 02:38:54 (removed)
[v4] 2018-07-13 01:45:53

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