Problems & Puzzles: Conjectures

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)