Application of theorems on primes to diophantine problems of a special type

In this paper we consider the problem of whether the equation
$$ n=\frac{\nu_1\varphi_1-\nu_2\varphi_2}{\nu_1-\nu_2}\qquad (\nu_1\ne\nu_2) $$
can be solved and of a lower bound for the number of solutions, subject to certain constraints on the density of the numbers $\nu$ and the distribution of the numbers $\varphi$ in arithmetic progressions.