(Compare also [@nakayama; @Webb:1970; @Li:1981; @Yang:1982] for some weaker results).

The conjecture was verified for all $n {\leqslant}10^{14}$ in [@swett]. In Table \[table:history\] we list a more complete history of these computations, but there may be further unpublished computations as well.

[\[table:history\]]{}

                   $5000$   ${\leqslant}$ 1950 
  ----------------------- -------------------- --
                   $8000$                 1962 
                  $20000$   ${\leqslant}$ 1969 
                 $106128$               1948/9 
                 $141648$                 1954 
                   $10^7$                 1964 
        $1.1 \times 10^7$                 1976 
                   $10^8$                 1971 
                   $10^9$                 1994 
                $10^{10}$                 1995 
    $1.6 \times  10^{11}$                 1996 
                $10^{10}$                 1999 
                $10^{14}$                 1999 
       $2\times  10^{14}$                 2012 

  : Numerical verifications of the Erdős-Straus conjecture. It appears that Terzi’s set of soluble residue classes is correct, but that the set of checked primes in these classes is incomplete. Another reference to a calculation up to $10^8$ due to N. Franceschine III (1978) (see [@guy; @erdosandgraham] and frequently restated elsewhere) only mentions Terzi’s calculation, but is not an independent verification. We are grateful to I. Kotsireas for confirming this (private communication).

Most of these previous approaches concentrated on the question whether $f(n)>0$ or not. In this paper we will instead study the average growth or extremal values of $f(n)$.

Since we clearly have $f(nm) {\geqslant}f(n)$ for any $n,m \in {\mathbb{N}}$, we see that to prove the Erdős-Straus conjecture it 