Define $R = \Bbb{Z}[X_1, X_2, \dots]$. Then place on $R$ the relations $$ X_1 + 1 = X_2, \\ X_2 + 2 = X_3, \\ X_3 + 2 = X_4, \\ X_5 + 2 = X_6, \dots \\ X_{2k-1} + 2 = X_{2k}, \ \forall k \geq 3 $$.
In other words, force the existence of the primes as we know them ($2 + 1 = 3, 3 + 2 = 5, 5+2 = 7, 11 + 2 = 13, \dots)$, making particularly sure there are infinitely many twin prime pairs $X_k, X_{k+1}$. We can accomplish this by taking the quotient of $R$ with the ideal generated by: $$\{ X_1 - X_2 + 1, \\ X_2 - X_3 + 2, \\ X_3 - X_4 + 2, \\ X_{2k-1} - X_{2k} + 2, \\ \forall k \geq 3\}$$
Now if you prove that the ring $R/I$ is isomorphic to $\Bbb{Z}$ then I think that proves the twin prime conjecture. If you prove that it's isomorphic to a ring containing $\Bbb{Z}$ then that may be of some interest as well.
So can you compute the quotient $R/I$ in some way?
Your quotient is $\,\cong\Bbb Z[X_1,\,X_5,X_7,X_9,X_{11},X_{13},\cdots]$. This is because in your quotient ring, $X_{2k}$ can be written in terms of $X_{2k-1}$ for $k\ge3$, and $X_4$ can be written in terms of $X_3$ which can be written in terms of $X_2$ which can be written in terms of $X_1$, and no other relations are imposed.
Notice also that primes are not even present in the definition of your quotient ring, so I see no connection between it and primes (let alone the twin prime conjecture).