$n(1-z_0) = t(p_2-p_1)$ is the constraint. In fact, this has some relationship with the twin primes conjecture
Starting with solving the equation $p_1x+p_2y=1$
If p1,p2 are twin primes and n is even, then the equation has solutions $(1,1)$ and solves the goldbach's conjecture
For the general goldbach conjecture which ask if every integer can be written as the sum of two primes, this only occurs if the prime gap is a multiple of the given integer
So the whole question resolves to, is there an integer for every gap between primes