Large gaps between consecutive prime numbers

THEOREM 1. Let R > 0. Then for any sufficiently large X, there are at least log X log2 X log4 X R (log3X)2 consecutive composite natural numbers not exceeding X. In other words, we have log X log2 X log4 X G(X)Zf(X) (log3*)2 for some function f(X) that goes to infinity as X —» oo. Theorem 1 settles in the affirmative a long-standing ... ................
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