more than ten important conjectures in prime numbers

Using proof by contradiction together with the Prime Number Theorem π(n) ∼ n lnn for sufficiently large n, we establish a general theorem on the distribution of prime numbers. Specifically, we prove that for every positive number h > 0, no matter how small, the interval [n, n +A(h).(lnn)1+h] always contains at least one prime number when n is sufficiently large.Here, A(h) is less than positive infinity and is a fixed constant for each fixed h > 0. This result confirms the validity of several classical conjectures on prime gaps, including Bertrand’s Postulate, Legendre’s Conjecture, Andrica’s Conjecture, Firoozbakht’s Conjecture, Oppermann’s Conjecture, and Brocard’s Conjecture, among others. Furthermore, we discuss the existence of large composite intervals such as [n! + 2, n! + n], which contain no prime numbers. The findings together provide a unified perspective on the upper bounds of prime gaps and the asymptotic behavior of primes within short intervals.