A200503 - OEIS (original) (raw)

(1) Conjectured upper bound: gaps between prime sextuplets (p, p+4, p+6, p+10, p+12, p+16) are smaller than 0.058*(log p)^7, where p is the prime at the end of the gap.

(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a(log(p/a)-1/3), where a = 0.058*(log p)^6 is the average gap between sextuplets near p, as predicted by the Hardy-Littlewood k-tuple conjecture.

Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of maximal gaps. Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.058 is reciprocal to the Hardy-Littlewood 6-tuple constant 17.2986...