# Find a sequence of whole numbers $n _ { 1 } , n _ { 2 } , \ldots$ such that $n _ { i - 1 } | n _ { i }$ for all $i \geq 2$

Problem : Find a sequence of whole numbers $$n _ { 1 } , n _ { 2 } , \ldots$$ such that $$n _ { i - 1 } | n _ { i }$$ for all $$i \geq 2$$ and for every $$k \in \mathbb{N}$$ there exists $$i$$ such that $$k | n _ { i }$$.

The subject of the exercice is "the splitting field of $$\mathbb{F}_p$$", with $$p$$ a prime. But this first question doesn't seem to relate (yet) to rings and fields theory.

This question makes me think of the proof of the the fundamental theorem of finite abelian groups with invariant factors.

• What have you tried? For instance, one famous sequence in particular stands out to me as almost made for this purpose. Try constructing such a sequence yourself, and maybe you can figure out which one I'm talking about. – Arthur May 12 at 17:28
• @arthur the Fibonacci sequence? – NotAbelianGroup May 12 at 17:30
• The Fibonacci sequence has, somewhere close to the beginning, a $2$ and then a $3$. Do we have $2\mid 3$? – Arthur May 12 at 17:31
• @Arthur Sorry that was a dumb and quick answer, you said "famous sequence" and I directly thought about this. I'm gonna look into that. – NotAbelianGroup May 12 at 17:33
• They will work wonderfully. You do have to check this yourself, though. You shouldn't trust strangers on the internet, after all :) – Arthur May 12 at 17:43

What about $$n_i = i!$$?