Erdős Problem #854

declared status 'open'. Formalized: no. 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 $a_{j+1}-a_j$? Current best: Erd\H{o}s 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 $k$. Prize: no. OEIS: A048670, A389839. Tags: number theory.

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