SOLVED

Are there infinitely many $n$ such that the largest prime factor of $n$ is $<n^{1/2}$ and the largest prime factor of $n+1$ is $<(n+1)^{1/2}$?

Pomerance has observed that if we replace $1/2$ in the exponent by $1/\sqrt{e}-\epsilon$ for any $\epsilon>0$ then this is true for density reasons (since the density of integers $n$ whose greatest prime factor is $\leq n^{1/\sqrt{e}}$ is $1/2$).

Steinerberger has pointed out this problem has a trivial solution: take $n=m^2-1$, and then it is obvious that the largest prime factor of $n$ is $\leq m+1\ll n^{1/2}$ and the largest prime factor of $n+1$ is $\leq m\ll (n+1)^{1/2}$ (these $\ll$ can be replaced by $<$ if we choose $m$ such that $m,m+1$ are both composite).

Given that Erdős and Graham describe the above observation of Pomerance and explicitly say about this problem that 'we know very little about this', it is strange that such a trivial obstruction was overlooked. Perhaps the problem they intended was subtly different, and the problem in this form was the result of a typographical error, but I have no good guess what was intended here.

See also [369].