becomes too great to handle.

To prove Theorem \[main\], we first use some solvability criteria for Type I and Type II solutions to obtain more tractable expressions for $f_{{\operatorname{I}}}(p)$ and $f_{{\operatorname{II}}}(p)$. As we shall see, $f_{{\operatorname{I}}}(p)$ is essentially (up to a factor of two) the number of quadruples $(a,c,d,f) \in {\mathbb{N}}^4$ with $4acd = p+f$, $f$ dividing $4a^2d+1$, and $acd {\leqslant}3p/4$, while $f_{{\operatorname{II}}}(p)$ is essentially the number of quadruples $(a,c,d,e) \in {\mathbb{N}}^4$ with $4acde = p + 4a^2d + e$ and $acde {\leqslant}3p/2$. (We will systematically review the various known representations of Type I and Type II solutions in Section \[represent-sec\].) This, combined with standard tools from analytic number theory such as the Brun-Titchmarsh inequality and the Bombieri-Vinogradov inequality, already gives most of Theorem \[main\]. The most difficult bound is the upper bounds on $f_{{\operatorname{I}}}$, which eventually require an upper bound for expressions of the form $$\sum_{a {\leqslant}A} \sum_{b {\leqslant}B} \tau( kab^2 + 1 )$$ for various $A,B,k$, where $\tau(n) := \sum_{d \mid n} 1$ is the number of divisors of $n$, and $d \mid n$ denotes the assertion that $d$ divides $n$. By using an argument of Erdős [@erdos], we obtain the following bound on this quantity:

\[kab\] For 