## On the Firoozbakht’s and Cramér’s Conjectures as N→∞

**Authors:** Reza Farhadian

Let p_n be the nth prime number. We prove that p_(n+1)<〖p_n〗^((n+1)/n ((logp_(n+1))/(logp_n )) ) for every n≥1. This inequality is weaker than the Firoozbakht’s conjecture p_(n+1)<〖p_n〗^((n+1)/n).Afterward we prove that the new inequality is equivalent to Firoozbakht’s conjecture, as n→∞, and hence the Cramér’s conjecture p_(n+1)-p_n=O(log^2 p_n ) to be hold, because the Firoozbakht’s conjecture is stronger than the Cramér’s conjecture, see [5].

**Comments:** 5 Pages.

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### Submission history

[v1] 2017-02-26 13:00:05

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