Let $p_1, p_2. ...,p_{x-1}, p_x$ be all of the the prime numbers from $2$ upto some arbitrarily chosen prime number $p_x$.
And consider the representation of a natural number $n$ in the form $(a_x, a_{x-1}, ..., a_2, a_1)$ where $a_y \equiv n \mod p_y$.
If $n$ is a prime,
$ n \equiv 0 \mod p_z$ iff $n = p_z$
$ n \equiv a_z \mod p_z : a_z < n $ iff $p_z \leq n$
$ n \equiv a_z \mod p_z: a_z = n$ iff $p_z > n$
My question is;
If we choose some arbitrary value of $x$, to give us a sort of base length for this representation of natural numbers in the way described above, and we chose random "digits", i.e we set some "random" congruence conditions; is it guaranteed that at least one prime number is the solution of our random congruence conditions if we make the rule that no "digit" can be $0$?.
For example, if I look at all numbers that are representable as (2,3,2,3,2,3,2,1), is there guaranteed to be a prime number that satisfies the respective system of congruence relations?
Obviously, we can use the Chinese remainder theorem to find solutions, but is there a proof that "one of the solutions must necessarily be a prime number" is True or False?