subset of one of the following residue classes:

-   $-e \mod 4ab$, where $a,b,e \in {\mathbb{N}}$ are such that $e \mid a+b$ and $(e,4ab)=1$. [@aigner]

-   $-4a^2d \mod f$, where $a,d,f \in {\mathbb{N}}$ are such that $4ad \mid f+1$. [@vaughan], [@rosati]

-   $-4a^2d-e \mod 4ade$, where $a,d,e \in {\mathbb{N}}$ are such that $(4ad,e)=1$. [@nakayama]

As indicated by the citations, many of these residue classes were observed to be solvable by polynomials in previous literature, but some of the conditions listed here appear to be new, and they form the complete list of all such classes. We prove Proposition \[res-class\] in Section \[solution\].

The results in this paper would also extend (with minor changes) to the more general situation in which the numerator $4$ in is replaced by some other fixed positive integer, a situation considered first by Sierpiński and Schinzel (see e.g. [@sierpinski; @vaughan; @Palama:1958; @Palama:1959; @Stewart:1964]).

We will not detail all of these extensions here but in Section \[lower-3\] we extend our study of the average number of solutions to the more general question on sums of $k$ unit fractions $${\label{m/nsumofk}}
\frac{m}{n}=\frac{1}{t_1} + \frac{1}{t_2} + \cdots +
\frac{1}{t_k}.$$ If $m {\leqslant}k$ the greedy algorithm (in this case also known 