Several questions in number theory deals with primes. For example, the twin prime conjecture states that $2=p_{n+1}-p_{n}$ infinitely many times. I know that $(p_{n})_{n≥1}$ is a sequence. But I am asking if one can consider this problem as a Diophantine equation (http://mathworld.wolfram.com/DiophantineEquation.html). The same question is for the Goldbach's conjecture (https://en.wikipedia.org/wiki/Goldbach%27s_conjecture).
In other words: Consider the equation: $$x-y-2=0.....(*)$$
Then the twin prime conjecture is true if and only if this equation has infinitely many pairs of primes solutions in $x$ and $y$.
More generally, I am asking if there exist a problem related to primes which is not transformable to a Diophantine equation.