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 Therefore conjecture 7 must be true.


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 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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