FALSIFIABLE
Open, but could be disproved with a finite counterexample.
Let $[1,\ldots,n]$ denote the least common multiple of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$,\[[1,\ldots,p_{k+1}-1]< p_k[1,\ldots,p_k]?\]
Erdős and Graham write this is 'almost certainly' true, but the proof is beyond our ability, for (at least) two reasons:
- Firstly, one has to rule out the possibility of many primes $q$ such that $p_k<q^2<p_{k+1}$. There should be at most one such $q$, which would follow from $p_{k+1}-p_k<p_k^{1/2}$, which is essentially the notorious Legendre's conjecture.
- The small primes also cause trouble.
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This page was last edited 07 October 2025.
Additional thanks to: Zachary Chase 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 #458, https://www.erdosproblems.com/458, accessed 2026-01-17
