colonel@sunybcs.UUCP (George Sicherman) (11/21/83)
Computer trivia question: what's the least positive integer n such that the binary expansion of 1/n has more 1's than 0's?
bob@burdvax.UUCP (11/23/83)
The least positive integer, n, for which the binary expansion of 1/n
contains more 1's than 0's is:
1 = 0.11111111 . . .bob@burdvax.UUCP (11/28/83)
The least positive integer n for which the non-trivial binary expansion of 1/n contains more 1's than 0's is 187. The binary expansion of 1/187 has a 40-bit period containing 21 1's and 19 0's. The table below shows all values of n < 1000 which share this property. This was achieved by brute force investigation (with grateful acknowledgement to the Vax 11/780). Does anyone have a generating function for such numbers? n period 1's 0's --- ------ --- --- 187 40 19 21 323 72 35 37 374 40 19 21 427 60 28 32 549 60 29 31 559 84 41 43 646 72 35 37 687 76 37 39 721 51 25 26 748 40 19 21 779 180 85 95 781 70 32 38 854 60 28 32 927 102 50 52 937 117 58 59 965 96 47 49 973 138 65 73
ka@hou3c.UUCP (Kenneth Almquist) (11/30/83)
The question does not specificly say that leading and trailing zeros are to be counted. If they are not counted, then the answer is n = 1. If leading and trailing zeros *are* to be included, then the number of zero digits will be infinite for any n. This means that the number of zero digits will be equal to the total number of digits. Since the number of one's digits cannot be greater than the total number of digits, the number of one's digits cannot be greater than the number of zero digits for any n. The preceding paragraph may be a little dense, but the important point is that the concept of greater than is somewhat nonintuitive when infinite quantities are involved. Kenneth Almquist
thomas@utah-gr.UUCP (Spencer W. Thomas) (12/01/83)
It was "obvious" to me that what the original question meant was 1. Consider all and only the digits following the binary point (including trailing zeros). 2. It was really asking for the first fraction where the *ratio* of 1s to 0s was greater than 1. =Spencer