instance $$f(1)=0, f(2)=3, f(3) = 12, f(4) = 10, f(5)=12, f(6)=39, f(7)=36, 
f(8)=46,  \ldots$$ We plot the values of $f(n)$ for $n {\leqslant}1000$, and separately restricting to primes $p{\leqslant}1000$ in Figures \[fig1\], \[fig2\].

![The value $f(n)$ for all $n {\leqslant}1000$.[]{data-label="fig1"}](f_all1000_11pt.pdf)

![The value $f(p)$ for all primes $p {\leqslant}1000$.[]{data-label="fig2"}](f_prime1000_11pt.pdf)

From these graphs one might be tempted to draw conclusions, such as “$f(n) \gg n$ infinitely often”, that we will refute in our investigations below.

The *Erdős-Straus conjecture* (see e.g. [@guy]) asserts that $f(n) >
0$ for all $n {\geqslant}2$; it remains unresolved, although there are a number of partial results. The earliest references to this conjecture are papers by Erdős [@Erdos:1950] and Obláth [@Oblath:1950], and we draw attention to the fact that the latter paper was submitted in 1948.

Most subsequent approaches list parametric solutions, which solve the conjecture for $n$ lying in certain residue classes. These soluble classes are either used for analytic approaches via a sieve method, or for computational verifications. For instance, it was shown by Vaughan [@vaughan] that the number of $n < N$ for which $f(n)=0$ is at most $N \exp( - c \log^{2/3} N )$ for some absolute constant $c>0$ and all sufficiently large $N$. 