The One Four Puzzle (?!)

I spend a couple hours a week with Mel Hochster, chair of the math department here at UMich, for the very serious purpose of doing cryptic crossword puzzles, a shared interest.  We almost never discuss actual math problems, but last week was an exception.  He says this is something he thought of a while back but never made much progress.

Suppose a set of positive integers contains the number 4 and is closed under the maps n\mapsto n! and n\mapsto \lfloor\sqrt{n}\rfloor.  Must the set be all of $\mathbb{Z}^+$?

The motivation comes from that game that almost everyone who likes to play with numbers has played, the four fours game, where you try to express various numbers in terms of four 4s and whatever operations you may wish.

This is the corresponding game with only one 4; accordingly, we only use unary operations, and only those which return integers.  What numbers can be expressed by starting with a 4, using factorials to go up, floored square roots to come down, going back and forth between these as desired.  Mel Hochster has verified by computer that all positive integers up to 1000 are so expressible and conjectures that all positive integers are.

But how to prove it.  He says that he asked Conway, who said something like “It’s hopeless.”  Not encouraging, sure, but I’m a person who spends time working on the Collatz problem, about which Erdős famously said “Mathematics is not ready for such problems.”

Apparently I have no idea when I’m out of my depth.  But it seems an interesting puzzle.  A good place to start seems to be a rather easier question.

Let n be a positive integer.  Let F(n) be the minimal m such that \left\lfloor (m!)^{1/2^{-k}}\right\rfloor = n for some integer k.  Can we get any nontrivial estimates on how F(n) grows?

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