"Proof 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.

结合反证法与素数定理（Prime Number Theorem）——即对于充分大的正整数n，素数计数函数π(n) ~ n/lnn——我们建立了一条关于素数分布的一般性定理。具体而言，我们证明了：对于任意小的正实数h>0，当n充分大时，区间[n, n + A(h)·(ln n)^(1+h)]内必存在至少一个素数。其中，A(h)为有限正常数，且对于固定的h>0，A(h)为定值。该结果验证了多个经典素数间隙猜想的正确性，包括伯特兰公设（Bertrand’s Postulate）、勒让德猜想（Legendre’s Conjecture）、安德丽卡猜想（Andrica’s Conjecture）、菲罗兹巴克特猜想（Firoozbakht’s Conjecture）、奥佩尔曼猜想（Oppermann’s Conjecture）与布罗卡德猜想（Brocard’s Conjecture）等。此外，我们还讨论了形如[n! + 2, n! + n]的无素数大合数区间的存在性。本研究成果为素数间隙的上界估计与短区间内素数的渐近行为提供了统一的理论视角。