Erdős problem 394

For the least $t_k(n)$ with $n \mid t_k(n)(t_k(n)+1)\cdots(t_k(n)+k-1)$ , do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with $c = 1/2048$ admissible in the $t_2$ bound.

Sources

The source's own material, retained exactly.

10 retained statements36c6d0fef71a

FormalConjectures/ErdosProblems/

394.lean

Retained formal statement4 of 9

They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$ , $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$ .

FormalConjectures/ErdosProblems/394.leanErdos394.erdos_394.variants.factorial_gap_conjecture1 lineExact file

1True ↔ {n | ∀ (k : ℕ), 2 ≤ k → k < n → Erdos394.t k n.factorial < Erdos394.t (k - 1) n.factorial - 1}.Infinite

OpenStatement only, no proof