Conference rusure::math

527.0. "rotational multiples" by ROXIE::OSMAN (and silos to fill before I feep, and silos to fill before I feep) Wed Jul 02 1986 20:40

Consider a number like:

	n = 0212765957446808510638297872340425531914893617

Numbers like this are interesting, because:

n * 10 = 2127659574468085106382978723404255319148936170
n *  6 = 1276595744680851063829787234042553191489361702
n * 13 = 2765957446808510638297872340425531914893617021
n * 36 = 7659574468085106382978723404255319148936170212
n * 31 = 6595744680851063829787234042553191489361702127
n * 28 = 5957446808510638297872340425531914893617021276
n * 45 = 9574468085106382978723404255319148936170212765
n * 27 = 5744680851063829787234042553191489361702127659
n * 35 = 7446808510638297872340425531914893617021276595
n * 21 = 4468085106382978723404255319148936170212765957
n * 22 = 4680851063829787234042553191489361702127659574
n * 32 = 6808510638297872340425531914893617021276595744
n * 38 = 8085106382978723404255319148936170212765957446
n *  4 = 0851063829787234042553191489361702127659574468
n * 40 = 8510638297872340425531914893617021276595744680
n * 24 = 5106382978723404255319148936170212765957446808
n *  5 = 1063829787234042553191489361702127659574468085
n *  3 = 0638297872340425531914893617021276595744680851
n * 30 = 6382978723404255319148936170212765957446808510
n * 18 = 3829787234042553191489361702127659574468085106
n * 39 = 8297872340425531914893617021276595744680851063
n * 14 = 2978723404255319148936170212765957446808510638
n * 46 = 9787234042553191489361702127659574468085106382
n * 37 = 7872340425531914893617021276595744680851063829
n * 41 = 8723404255319148936170212765957446808510638297
n * 34 = 7234042553191489361702127659574468085106382978
n * 11 = 2340425531914893617021276595744680851063829787
n * 16 = 3404255319148936170212765957446808510638297872
n * 19 = 4042553191489361702127659574468085106382978723
n *  2 = 0425531914893617021276595744680851063829787234
n * 20 = 4255319148936170212765957446808510638297872340
n * 12 = 2553191489361702127659574468085106382978723404
n * 26 = 5531914893617021276595744680851063829787234042
n * 25 = 5319148936170212765957446808510638297872340425
n * 15 = 3191489361702127659574468085106382978723404255
n *  9 = 1914893617021276595744680851063829787234042553
n * 43 = 9148936170212765957446808510638297872340425531
n *  7 = 1489361702127659574468085106382978723404255319
n * 23 = 4893617021276595744680851063829787234042553191
n * 42 = 8936170212765957446808510638297872340425531914
n * 44 = 9361702127659574468085106382978723404255319148
n * 17 = 3617021276595744680851063829787234042553191489
n * 29 = 6170212765957446808510638297872340425531914893
n *  8 = 1702127659574468085106382978723404255319148936
n * 33 = 7021276595744680851063829787234042553191489361
n *  1 = 0212765957446808510638297872340425531914893617


Define "like".

/Eric

    The number in question is the repeating fractional part of 1/47
    multiplied by a large enough power of 10 to make it an integer.
    This works for 1/p for p= primes > 5, if the period of the repeating
    decimal fraction is of length p-1. Thus it works for 1/7 but fails
    for 1/13.
    
    The repeating period of 1/19, as I recall, has the interesting property
    that if you write down the period multiplied by 1, 2, ..., 18 you
    will form an 18x18 magic square of single digits.

    Isn't the magic square property necessarily true of ALL such "long"
    1/p fractions ?  For instance, look back at my 1/47 example.  Each
    row certainly adds to the same thing, since the digits are the
    same just rotated.  Each column also has the same digits.  Hence
    it's a magic square, except perhaps for the diagonals, which are
    probably fine too, and if not, can be made fine by appropriate
    reordering of the rows which, fortunately, won't affect the integrity
    of the columns.
    

    For 1/19, no reordering is required for the diagonals to add up
    correctly, and that is what makes the square magic. For 1/7, for
    a counterexample, the array of 1...n-1 multiples is
    	142857
    	285714
    	428571
    	571428
    	714285
    	857142
    and the diagonal sums are 23 and 31.

    
    re .2:
    
    I believe one of the major conditions of "magicity" in a magic square
    is that none of the elements repeats.  There certainly seems to
    be a certain lack of magic in the square
    
    				2 2 2
    				2 2 2
    				2 2 2
    

> Consider a number like:
>
>    n = 0212765957446808510638297872340425531914893617
>
> Numbers like this are interesting, because:

    Eric, surely you know that _all_ numbers are interesting!
    
    In fact, even the magic square in .4 holds a certain amount of interest.

      John $-}

    Actually, John is correct.  All numbers are interesting.  Here's
    a proof:
    
    Suppose there existed a set of uninteresting numbers.  Call it S.
    
    		S = {u1, u2, u3 . . . un}
    
    There is some minimum number in this set.  Call it Sm.
    
    		Sm = Sj (for some 1 <= j <= n)
    
    This Sm is the smallest uninteresting number.
    
    Hence Sm is of some interest.
    
    Operating by induction, we reduce our set to null.
    
    /Eric

    
    re .-1
    
>    Operating by induction, we reduce our set to null.
    
    Unfortunately, the case n = 1 is false. The smallest element of a set
    with one value is not interesting.
    
    
    Can anyone generalise this proof to real numbers? (in which case a set
    of uninteresting numbers may not even have a smallest element)