Erdős Problem #5

The normalised prime gap \(\frac{p_{n+1}-p_n}{\log n}\), where \(p_n\) denotes the \(n\)-th prime.

The set \(S\) of limit points of \(\frac{p_{n+1}-p_n}{\log n}\). Only the *finite* limit points are collected here; that \(\infty\) is also a limit point is Westzynthius' theorem, recorded separately as `erdos_5.variants.westzynthius`. Erdős' question, as well as [HiMa88] and [Pi16], normalises the prime gaps by \(\log n\), whereas [GPY09], [BFM16] and [Me20] normalise by \(\log p_n\). Since \(\log p_n/\log n \to 1\) the two normalisations have the same limit points, so all the results below are stated for