Very Large Gaps between Consecutive Primes, Original Research Article

LetG(X) denote the largest gap between consecutive primes belowX. Improving earlier results of Erdimages, Rankin, Schönhage, and Maier-Pomerance, we proveG(X)greater-or-equal, slanted(2eγ+o(1)) log Xlog2 Xlog4 X(log3 X)−2,where logν Xdenotes theν-fold iterated logarithm function andγis Eulerʹs constant. The new tool used is a combinatorial result proved by probabilistic methods.