Elementary Primes Counting Methods

This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the subset of Germain primes PG = {p and 2p + 1 are primes}, and the subset of quadratic primes Pf = {p = n^2 + 1 primes}. These subsets of primes are widely believed to be infinite subsets of prime numbers.