Problems & Puzzles: Conjectures

Conjecture 7. The Cramer's Conjecture

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

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

Later Cramer assumed the Riemann Hypothesis and showed that dn = O(p1/2n.Lnpn), 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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