to prove that even in the prime case the average number of solutions grows quickly, when $k$ increases.

We will focus on the case of *Type II solutions*, in which $t_2,\ldots,t_k$ are divisible by $n$. The classification of solutions that we give below also works for other divisibility patterns, but Type II solutions are the easiest to count, and so we shall restrict our attention to this case. Strictly speaking, the definition of a Type II solution here is slightly different from that discussed previously, because we do not require that $t_1$ is coprime to $n$. However, this coprimality is automatic when $n$ is prime (otherwise the right-hand side of would only be at most $k/n$). For composite $n$, it is possible to insert this condition and still obtain the lower bound , but this would complicate the argument slightly and we have chosen not to do so here.

For given $m,k,n$, let $f_{m,k,{{\operatorname{II}}}}(n)$ denote the number of Type II solutions. Our main result regarding this quantity is the following lower bound on this quantity:

\[many-solutions\] Let $m > k{\geqslant}3$ be fixed. Then, for $N$ sufficiently large, one has $$\label{many-1}
\sum_{n {\leqslant}N} f_{m,k,{{\operatorname{II}}}}(n) \gg_{m,k} N (\log N)^{2^{k-1}-1}$$ and $$\label{many-2}
\sum_{p 