prime numbers

yes

It has been proved that there are infinitely many twin primes (primes a and b with a difference 2, i.e., a - b = 2).

Proof of the Twin Primes Conjecture Using Euclid's Logic:


Suppose that i(1)=i(2)=i(3) . . . =i(r)=2 are all the twin prime intervals. Let I(r)=2^r 1, i(r 1)=2 and I(r 1)=2^(r 1) 1 from which it follows that 2I(r)=I(r 1) 1. For all values of r, since interval i(r 1) divides 2I(r) (without dividing I(r) and I(r 1),) it is yet another interval and intervals i(1)=i(2)=i(3)= . . . =i(r) are not all the twin prime intervals, i.e., r->infinite in i(1)=i(2)=i(3)= . . . =i(r)=2.