(thus $|A \cap \{1,\ldots,N\}|/N \to 1$ as $N \to \infty$).

Finally, one has $$f(p){\geqslant}\exp \left( (\frac{\log 3}{2} - o(1)) \log \log p \right) \gg (\log p)^{0.549}$$ for all primes $p$ in a subset $B$ of primes of relative density $1$ (thus $|\{ p \in B: p {\leqslant}N\}|/{|\{ p: p {\leqslant}N \}|} \to 1$ as $N \to \infty$).

As the proof shows the first two lower bounds are already valid for sums of two unit fractions. The result directly follow from the growth of certain divisor functions. An even better model for $f(n)$ is a suitable superposition of several divisor functions. The proof will be in Section \[lowerboundsec-2\].

Finally, we consider (following [@mordell], [@schinzel]) the question of finding polynomial solutions to . Let us call a primitive residue class $n = r \mod q$ *solvable by polynomials* if there exist polynomials $P_1(n), P_2(n), P_3(n)$ which take positive integer values for all sufficiently large $n$ in this residue class (so in particular, the coefficients of $P_1,P_2,P_3$ are rational), and such that $$\frac{4}{n} = \frac{1}{P_1(n)} + \frac{1}{P_2(n)} + \frac{1}{P_3(n)}$$ for all $n$. Here we recall that a residue class $r \mod q$ is *primitive* if $r$ is coprime to $q$. One could also 