OPEN
This is open, and cannot be resolved with a finite computation.
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 $a_{j+1}-a_j$?
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 $j$.
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This page was last edited 04 November 2025.
Additional thanks to: Dogmachine, Wouter van Doorn, and Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #854, https://www.erdosproblems.com/854, accessed 2026-01-16
