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Re: Challenge: Chasing Knuth's Conjectureby tall_man (Parson) 
on Mar 29, 2005 at 21:33 UTC ( #443270=note: print w/ replies, xml )  Need Help?? 
More recently, we have Knuth's Conjecture: "Representing Numbers Using Only One 4", Donald Knuth, (Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp.308310). 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:
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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