note
tall_man
On [http://www.magicdragon.com/math.html|this page] there is a reference to Knuth's conjecture but it says he starts with 4:
<p><strong>
More recently, we have Knuth's Conjecture:
<p>"Representing Numbers Using Only One 4", Donald Knuth,
(Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp.308-310).
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.
<p>For example: Knuth shows how to make the number 64 using only one 4:</strong>
<code>
|_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt
|_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt
|_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt
|_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt
|_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt
|_ sqrt |_ sqrt |_ sqrt sqrt sqrt sqrt sqrt
(4!)! _| ! _| ! _| ! _| ! _| ! _| ! _| ! _|
</code>
<p><strong>As to notation in the above example, he means sqrt n! stands for
sqrt (n!), not (sqrt n)!
<p>Knuth further points out that |_ sqrt |_ X _| _| = |_ sqrt X _|
so that the floor function's brackets are only needed around the entire
result and before factorials are taken.
<p>He CONJECTURES that all integers may be represented that way:
"It seems plausible that all positive integers possess such a
representation, but this fact (if true) seems to be tied up with very deep
propertis of the integers."
<p>Your Humble Webmaster believes that Knuth is right, for 9 as well as 4,
and will prove that in a forthcoming paper.
<p>Knuth comments: "The referee has suggested a stronger conjecture, that a
representation may be found in which all factorial operations precede all
square root operations; and, moreover, if the greatest integer function
(our floor function) is not used, an arbitrary positive real number can
probably be approximated as closely as desired in this manner."</strong>
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