Problem #375: Is it true that for any $n,k\geq 1$, if $n+1,\ldots,n+k$...

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Progress

Note this is trivial when $k\leq 2$. Originally conjectured by Grimm [Gr69]. This is a very difficult problem, since it in particular implies $p_{n+1}-p_n <p_n^{1/2-c}$ for some constant $c>0$, in particular resolving Legendre's conjecture.

Grimm proved that this is true if $k\ll \log n/\log\log n$. Erdős and Selfridge improved this to $k\leq (1+o(1))\log n$. Ramachandra, Shorey, and Tijdeman [RST75] have improved this to\[k\ll\left(\frac{\log n}{\log\log n}\right)^3.\]This is problem B32 in Guy's collection [Gu04].

See also [860].