The unity of the prime number theorem, Bertrand's postulate and Euler's proof

Abstract: In this paper, we present two novel approaches to prove the Polignac's conjecture. ⑴ For any integer n such that   is prime and  , we construct interval  . By synthesizing the prime number theorem, Bertrand's postulate and Euler's proof to analyze the gap between   and primes in the interval  , we prove the Polignac's conjecture. ⑵ For any integer n ≥ 2, we construct consecutive intervals   and  . By synthesizing Bertrand's postulate and the prime number theorem to analyze the gap of primes between the consecutive intervals, we prove the Polignac's conjecture.