Difference between consecutive primes

Suppose that B is a sufficiently large positive constant, ε is a sufficiently small positive constant, X and N are sufficiently large. It is mainly proved that i) for the positive integers n, X<n≤2X, except for O(X log-B X) values, the interval (n, n+n1/14+ε) contains a prime number; ii) if A = N1/2+s, then the even numbers in the interval (N, N+A), except for 0(Alog-B N) values, are all Goldbach numbers.

Suppose that B is a sufficiently large positive constant, ε is a sufficiently small positive constant, X and N are sufficiently large. It is mainly proved that i) for the positive integers n, X&lt;n≤2X, except for O(X log-B X) values, the interval (n, n+n1/14+ε) contains a prime number; ii) if A = N1/2+s, then the even numbers in the interval (N, N+A), except for 0(Alog-B N) values, are all Goldbach numbers.