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in reply to Challenge: Chasing Knuth's Conjecture

On this page there is a reference to Knuth's conjecture but it says he starts with 4:

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."