A question which Chalk and L. Moser asked me several years ago led me to the following problem: Let 0 < x ≤ y. Estimate the smallest f(x) so that there should exist integers u and v satisfying

1

I am going to prove that for every ∊ > 0 there exist arbitrarily large values of x satisfying

2

but that for a certain c > 0 and all x

3

A sharp estimation of f(x) seems to be a difficult problem. It is clear that f(p) = 2 for all primes p. I can prove that f(x)→ ∞ and f(x)/loglog x → 0 if we neglect a sequence of integers of density 0, but I will not give the proof here.