are motivated by the Erdős-Straus equation for prime denominator.

We prove this proposition in Section \[upper-sec\].

The main tools here are the multiple representations of Type I and Type II solutions available (see Section \[represent-sec\]) and the divisor bound . The values of $f(p)$ appear to fluctuate in some respects as the values of the divisor function. The average values of $f(p)$ behave much more regularly.

Moreover, in view of Theorem \[main\], one might also expect to have $f(n) \ll_{\varepsilon}n^{\varepsilon}$ for any ${\varepsilon}>0$, but such logarithmic-type bounds on solutions to Diophantine equations seem difficult to obtain in general (Proposition \[upper\] appears to be the limit of what one can obtain purely from the divisor bound alone).

In the reverse direction, we have the following lower bounds on $f(n)$ for various sets of $n$:

[\[lowerboundturankubilius\]]{} For infinitely many $n$, one has $$f(n){\geqslant}\exp ((\log 3 +o(1))\frac{\log n}{\log \log n}),$$ where $o(1)$ denotes a quantity that goes to zero as $n \to \infty$.

For any function $\xi(n)$ going to $+\infty$ as $n \to \infty$, one has $$f(n){\geqslant}\exp \left( \frac{\log 3}{2}\log \log n -
O(\xi(n) \sqrt{\log \log n})\right) \gg (\log n)^{0.549}$$ for all $n$ in a subset $A$ of natural numbers of density $1$ 