Let us consider the strong twin conjecture: For all positive integer $n$ there exist a prime $p$ such that $$n+4<p<2^n2^4$$ and $p$ is a prime and $p+2$ is a prime
Since the inequalities and the set of primes are both Diophantine, we can construct a polynomial $P(X)$ such that the strong twin conjecture is equivalent to the following statement:
For all positive integer $n$ there exist a prime $p$ such that $P(X)=0$ where $X$ is a vector of several variables.
Now, in a book by Matiyasevich's: https://mitpress.mit.edu/books/hilberts-10th-problem
The author claim that the strong twin conjecture can be transformed into the unsolvability of a particular Diophantine equation.
Then my question is: How one can find this Diophantine equation or I am asking about a reference containing this equation.