Infinitely many pairs of primes $p$ and $p+2$
We take the pre-sieved set to be all natural numbers $N=\{1,2,3,\dots\}$ with a sieve system:single sieve,double sieve,.... With single sieve, i.e. , remove out the multiple of a prime, we derive all the primes. With double sieve, i.e. , remove out the multiple and the multiple of a prime and $-2$ simultaneously, we get all the prime twins and prove that infinitely many prime twins exist under suitable conditions. Finally, with special 4 sieve, we prove that infinitely many prime twins exist without any restriction.
math.GM↗