"Knuth shows how (using a computer program he wrote) all integers from 1 through 207 may be represented with only one 4, varying numbers of square roots, varying numbers of factorials, and the floor function." So what are the current smallest numbers of which we don't know their Knuth's presentations? Warut On Sat, Sep 18, 2010 at 2:03 PM, Warut Roonguthai <warut822@gmail.com> wrote:
Here is the representation of 197 that I just found:
844 = [(44!!)^(1/2^185)] 197 = [(844!!)^(1/2^7003)].
I used simple Stirling's formula for large factorial approximations. An independent verification is needed here.
Warut
On Tue, Jul 20, 2010 at 3:58 AM, Erich Friedman <efriedma@stetson.edu> wrote:
this is actually a variant of an old puzzle, with 4 replaced by 3. they are equivalent problems since 4 can be done from 3 and 3 from 4.
every number up to 196 has such a representation. i'm guessing all numbers do.
erich
On Jul 19, 2010, at 4:12 PM, Marc LeBrun wrote:
Can every integer be represented by a composition of integer square root ($) and factorial (!) applied to 3?
Some example representations: 3 = 3 6 = 3! 1 = 3$ 2 = 3!$ 720 = 3!! 26 = 3!!$ 5 = 3!!$$ 120 = 3!!$$! 10 = 3!!$$! ...
If not, what is the smallest counterexample?
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