in reply to Challenge: Chasing Knuth's Conjecture
More recently, we have Knuth's Conjecture:
"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.
For example: Knuth shows how to make the number 64 using only one 4:
|_ 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!)! _| ! _| ! _| ! _| ! _| ! _| ! _| ! _|
As to notation in the above example, he means sqrt n! stands for
sqrt (n!), not (sqrt n)!
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.
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."
Your Humble Webmaster believes that Knuth is right, for 9 as well as 4,
and will prove that in a forthcoming paper.
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."
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Re^2: Challenge: Chasing Knuth's Conjecture
by Anonymous Monk on Mar 30, 2005 at 08:56 UTC |