How one can solves an equation of the form: $ap_{n}+bn=c$

Question

My question is: How can one solve an equation of the form:

$$ap_{n}+bn=c$$ where $p_{n}$ is the $n^{th}$ prime number, $a,b$ and $c$ are integers.

Answer 1

Let $k,m$ be any two integers and $r$ the remainder of $m/n$. Then rearranging your equation into $$a=\frac{c-bn}{p_n} $$ we see that all the triplets of solutions are of the form $(a,b,c)=\left(k,\displaystyle \frac{m-r}{n}, m-r+k p_n\right).$

This should be the best you will get, if you refute approximations.