All Random Solved Random Open
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 two reasons (at least):
  • 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.
Additional thanks to: Zachary Chase