p \log \log p)$ for all $p$ in a subset of the primes of relative density $1$. This may provide an explanation as to why analytic methods (such as the circle method) appear to be insufficient to resolve the Erdős-Straus conjecture, as such methods usually only give non-trivial lower bounds on the number of solutions to a Diophantine equation in the case when the number of such solutions grows polynomially with the height parameter $N$. (There are however some exceptions to this rule, such as Gallagher’s results [@gallagher] on representing integers as the sum of a prime and a bounded number of powers of two, but such results tend to require a large number of summands in order to compensate for possible logarithmic losses in the analysis.)

The double logarithmic factor $\log \log N$ in the above arguments arises from technical limitations to our method (and specifically, in the inefficient nature of the Brun-Titchmarsh inequality when applied to very short progressions), and we conjecture that it should be eliminated.

In view of these results, one can naively model $f(p)$ as a Poisson process with intensity at least $c \log^3 p$ for some absolute constant $c$. Using this probabilistic model as 