Consider a system of $n$ divisibility conditions on $n$ prime variables: $$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$ where $a_{i,j}$ are bounded integers. How many solutions are there with all $p_i$ prime, distinct and in a range $[N_0,N_1]$? (We can assume that $N_0$ is quite a bit larger than all $a_{i,j}$.) Or, put otherwise: under what conditions are there very few solutions?
Of course, some conditions are needed on $a_{i,j}$ for the system to have few solutions: if $a_{i,j}=0$ for all $(i,j)$, the system does have very many solutions!
Further thoughts: to make the question more precise, we may require each $p_i$ to be in a dyadic interval $\lbrack M_i, 2 M_i\rbrack$, and write $N_0$ for $\max_i M_i$. Can we hope (given $a_{i,j}$ not terribly degenerate) for a bound of the form $$\leq \frac{\prod_i O(M_i)}{N_0^n} \;\;\;\;\text{or, even better,} \leq \frac{\prod_i O(M_i/\log M_i)}{N_0^n}$$ on the total number of solutions? (See the comments below for an example suggesting that such a bound would be tight.)
If all $M_i$ are equal (or bounded by $O(N_0)$, say) then the divisibility conditions can be replaced by a system of equations $c_i p_i = a_{i,1} p_1 + \dotsc + a_{i,n} p_n$, $c_i$ bounded. That is a system of $n$ equations in $n$ variables, and thus should not in general have non-trivial solutions (meaning: it will not have solutions, as $0$ is not a prime). (Note in passing the case $c_i = \sum_{j} a_{i, j}$, which gives a solution with $p_i$ non-distinct, and in fact all the same; we are not counting such solutions, but it's good to keep them in mind.) So it is really the case of $M_i$ not all of the same size that is interesting.
Already a bound of the form $$\leq \frac{\prod_i O(M_i)}{N_0^{n/2}},$$ say, would be non-obvious and useful. An elementary solution could be particularly nice.