Can we use finite combination of factorials, surds and floor function on a combination of single number, say 4 to form every number?

Asked Jun 27 '21 at 16:48

Active Jun 28 '21 at 04:22

Viewed 108 times

So the title is much pretty much the question, one thing to note that is you strictly have to use factorials, surds and floor function.

Edit:

The problem was a little bit ambiguous earlier. So, here are more details:

You can use only one number $4$.
We can apply any number of factorials and nth roots, and floor functions. Also, if it's not possible with factorials, is it with gamma functions?

edited Jun 28 '21 at 04:22

asked Jun 27 '21 at 16:48

ThunderWolf

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Have you made any attempts to solve this problem? If so, where have you run into trouble? – Carl Schildkraut Jun 27 '21 at 16:51
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By surd do you mean the $n$-th root of a number (n>1)? That is, square root or cube root or etc. – coffeemath Jun 27 '21 at 16:56
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You certainly won't form non-computable or even non-algebraic numbers. – aschepler Jun 27 '21 at 16:56
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yeah, that definition. – ThunderWolf Jun 27 '21 at 16:57
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@ThunderWolf And also by "every number" do you just mean positive integers? [referring to comment by aschepler] – coffeemath Jun 27 '21 at 16:58
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Yes, positive inetgrers @coffeemath – ThunderWolf Jun 27 '21 at 17:04
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Of the top of my head, I don't see how you could start with $1$ or $2$ and create larger numbers. – Keith Backman Jun 27 '21 at 17:08
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Can one use the $4$ more than once? If so and we can get $1$ using all three of factorial, surd, and floor then that could just be repeated to get the other positive integers, unless you don't allow adding. – coffeemath Jun 27 '21 at 17:08
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No, @coffeemath, we can use 4 only once. Actually, this problem is self-made and inspired from the "Four Fours challenge", when we observed that factorials make numbers larger and larger, and surds make them smaller and smaller. (If you're wondering how I know this, the OP and I are friends and made this problem together.) – ultralegend5385 Jun 27 '21 at 23:13
This seems like more of a combinatorics question. – Spectre Jun 28 '21 at 02:12
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Very closely related: https://math.stackexchange.com/questions/24964/repeated-factorials-and-repeated-square-rooting – Gerry Myerson Jun 28 '21 at 04:59
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That link is helpful @GerryMyerson! – ultralegend5385 Jun 28 '21 at 05:30

Can we use finite combination of factorials, surds and floor function on a combination of single number, say 4 to form every number?

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