Problem #240: Is there an infinite set of primes $P$ such that if...

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Progress

Originally asked to Erdős by Wintner.

The limit is infinite for a finite set of primes, which follows from a theorem of Pólya [Po18], that if $f(n)$ is a quadratic integer polynomial without repeated roots then as $n\to \infty$ the largest prime factor of $f(n)$ also approaches infinity. Indeed, if $P$ is a finite set of primes and $(a_i)$ is the set of integers divisible only by primes in $P$, and $a_{i+1}-a_i$ is bounded, then there exists some $k$ such that $a_{i+1}=a_i+k$ infinitely often, which contradicts Pólya's theorem with $f(n)=n(n+k)$.

Tijdeman [Ti73] proved that, if $P$ is a finite set of primes, then\[a_{i+1}-a_i \gg \frac{a_i}{(\log a_i)^C}\]for some constant $C>0$ depending on $P$.

Tijdeman [Ti73] resolved this question, proving that, for any $\epsilon>0$, there exists an infinite set of primes $P$ such that, with $a_i$ defined as above,\[a_{i+1}-a_i \gg a_i^{1-\epsilon}.\]See also [368].