  The idea of generating prime numbers through sequence of sets of co-primes
was the starting point of this paper that ends up by proving two conjectures,
the existence of infinitely many twin primes and the Goldbach conjecture. The
main idea of our approach is summarized on the creation and on the analyzing
sequence of sets of distinct co-primes with the first $n$ primes, $\left\{ p_i
:\, i\leq n \right\}$, and the important properties of the modulus linear
combination of the co-prime sets, $H=\left(1,p_{n+1},..., \Pi_{i=1}^n
p_i-1\right) $, that gives sets of even numbers $\{0,2,4,..., \Pi_{i=1}^n p_i
-2 \}$. Furthermore, by generalizing our approach, the Polignac conjecture "the
existence of infinitely many cousin primes, $p_{n+1}-p_{n}=4$," and the
statement that "every even integer can be expressed as a difference of two
primes," are derived as well.
