the reader, we give the proof in Section \[odd-sec\]. A variant of this proposition was also established in [@bello]. Note that this does not disprove the Erdős-Straus conjecture, since the inequality does not hold with equality on perfect squares; but it does indicate a key difficulty in attacking this conjecture, in that when showing that $f_{{\operatorname{I}}}(p)$ or $f_{{\operatorname{II}}}(p)$ is non-zero, one can only use methods that *must necessarily fail* when $p$ is replaced by an odd square such as $p^2$, which already rules out many strategies (e.g. a finite set of covering congruence strategies, or the circle method).

Next, we establish some upper bounds on $f_{{\operatorname{I}}}(n), f_{{\operatorname{II}}}(n)$ for fixed $n$:

\[upper\] For any $n \in {\mathbb{N}}$, one has $$f_{{\operatorname{I}}}(n) \ll n^{3/5 + O(1/\log \log n)}$$ and $$f_{{\operatorname{II}}}(n) \ll n^{2/5 + O(1/\log \log n)}.$$ In particular, from this and one can conclude that for any prime $p$ one has $$f(p) \ll p^{3/5 + O(1/\log \log p)}.$$

This should be compared with the recent result in [@browning], which gives the bound $f(n) \ll_{\varepsilon}n^{2/3+{\varepsilon}}$ for all $n$ and all ${\varepsilon}>0$. For composite $n$ the treatment of parameters dividing $n$ appears to be more complicated and here we concentrate on those two cases that 