This is known now for $1\leq k\leq 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erdős [Er52]. The case $k=3$ was proved independently by Schlage-Puchta [ScPu06] and Friedlander, Luca, and Stoiciu [FLC07]. The case $k=4$ was proved by Pratt [Pr22].

This is discussed in problem B14 of Guy's collection [Gu04].

View the LaTeX source

This page was last edited 28 September 2025.

1 comment on this problem

• It is worth noting that Schlage-Puchta [ScPu06] proved that if Schinzel’s conjecture holds, then this is irrational for all \(k \in \mathbb{N}\). Friedlander, Luca, and Stoiciu [FLC07] further showed that if the Prime \(k\)-tuples Conjecture holds, then this is also irrational for all \(k \in \mathbb{N}\).

  Quanyu Tang — 14:20 on 05 Sep 2025

  👍 1 📝 1 🤖 0