d_n congruent p_(n + 1) - p_n. The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (OEIS A001223). Rankin has shown that d_n>(c ln n ln ln n ln ln ln ln n)/(ln ln ln n)^2 for infinitely many n and for some constant c. At a March 2003 meeting on elementary and analytic number in Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof that lim inf_(n->∞) (p_(n + 1) - p_n)/(ln p_n) = 0 (Montgomery 2003). Unfortunately, this proof turned out to be flawed.

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