Conjecture 7. The Cramer's Conjecture

Problems & Puzzles: Conjectures

Conjecture 7. The Cramer's Conjecture

Asking by the inferior limit for the "gap" between consecutive primes, namely, d_n = p_n+1 - p_n, one of the first conjectures became from Cramer, who conjectured in 1937 that d_n = O((Lnp_n)²), which means that

limit sup { (p_n+1 - p_n )/ (Lnp_n)²} =1 as n -> infinite

Later Cramer assumed the Riemann Hypothesis and showed that d_n = O(p^1/2_n.Lnp_n), but as the Riemann Hypothesis is another unproved conjecture, the two Cramer statements were and remain unproved.

(Ref. 1, p. 253 ; Ref. 2, p. 7)

On February 27, 2017, Reza Farhadian wrote:

I think that conjecture 7 is true. Because we know that the Firoozbakht's conjecture (or conjecture 30) is stronger than conjecture 7. On the other hand I proved that conjecture 30 is true as n goes to infinity (see http://vixra.org/abs/1702.0318). Therefore conjecture 7 must be true.

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On Dec 25, 2017 Reza Farhadian wrote:

"Already (Refer to previous post at the date February 27, 2017), I thought I was able to prove the Firoozbakht's conjecture by my article at the link http://vixra.org/abs/1702.0318. But recently I realized that this is not true. My article was published without arbitration, so it was not valid, hence I removed the article from the link."

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