{\leqslant}N} f_{m,k,{{\operatorname{II}}}}(p) \gg_{m,k} \frac{N (\log N)^{2^{k-1}-2}}{\log \log N}.$$

Our emphasis here is on the exponential growth of the exponent. In particular, as $k$ increases by one, the average number of solutions is roughly squared. The denominator of $\log \log N$ is present for technical reasons (due to use of the crude lower bound on the Euler totient function), and it is likely that it could be eliminated (much as it is in the $m=4,k=3$ case) with additional effort.

If we let $f_{m,k}(n)$ be the total number of solutions to (not just Type II solutions), then we of course obtain as a corollary that $$\sum_{n {\leqslant}N} f_{m,k}(n) \gg_k N (\log N)^{2^{k-1}-1}.$$ We do not expect the power of the logarithm to be sharp in this case (cf. Remark \[heath-remark\]). For instance, in [@huangandvaughan] it is shown that $$\sum_{n {\leqslant}N} f_{m,2}(n) = \left(\frac{1}{\phi(m)}+o(1)\right) N \log^2 N$$ for any fixed $m$.

Note that the equation can be rewritten as $$\frac{1}{mt_1} + \cdots + \frac{1}{mt_k} + \frac{1}{-n} = 0,$$ which is primitive when $n$ is prime. As a consequence, we obtain a lower bound for the number of integer points on the (generalised) Cayley surface:

Let $k{\geqslant}3$. The number of integer points of the 