Problem Statement
Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.
If $1=a_1<a_2<\cdots a_{\phi(n_k)}=n_k-1$ is the sequence of integers coprime to $n_k$, then estimate the smallest even integer not of the form $a_{i+1}-a_i$. Are there\[\gg \max_i (a_{i+1}-a_i)\]many even integers of the form $a_{j+1}-a_j$?
If $1=a_1<a_2<\cdots a_{\phi(n_k)}=n_k-1$ is the sequence of integers coprime to $n_k$, then estimate the smallest even integer not of the form $a_{i+1}-a_i$. Are there\[\gg \max_i (a_{i+1}-a_i)\]many even integers of the form $a_{j+1}-a_j$?
Categories:
Number Theory
Progress
This was asked by Erdős in Oberwolfach (most likely in 1986). Clearly all differences $a_{i+1}-a_i$ are even. Erdős first thought that (for large enough $k$) all even $t\leq \max(a_{i+1}-a_i)$ can be written as $t=a_{j+1}-a_j$ for some $j$, but in [Ob1] writes 'perhaps this is false', and reports some computations of Lacampagne and Selfridge that this fails for $n_k=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$ which 'show some doubt on [his] conjecture', and says it could fail for all or infinitely many $k$.In [Ob1] he also asks about the set of $j$ for which $a_{j+1}-a_j=\max(a_{i+1}-a_i)$, and in particular asks for estimates on the number of such $j$ and the minimal value of such a $j$.
Source: erdosproblems.com/854 | Last verified: January 19, 2026