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A presentation of the following article was given at the Joint Mathematics Meetings, January 10-13, 1996, Orlando, Florida.

Carousel Numbers:
A Lead-in to Number Theory

Gary Klatt
Mathematics and Computer Science Dept.
Univ. of Wisconsin - Whitewater
Whitewater, WI 53190
fax: (414) 472-5238
e-mail: klattg@uwwvax.uww.edu
phone: (414) 472-5173

    ------------------------------------------------------------------------
    0.     BACKGROUND/OVERVIEW
    ------------------------------------------------------------------------
    
    
    PROJECT FOR ALGEBRA AND NUMBER THEORY
    COURSE FOR PRE-SERVICE TEACHERS
    
      -    Ideal for discovery learning
      -    Students work in groups and individually
      -    Uses Derive and custom programs
      -    Students write up results in a paper
    
    
    
    FOUR DIVERSE PROBLEMS
    
      -    Each generates interesting questions and conjectures
      -    3 of the 4 are elementary
      -    Surprising connections to the main problem
      -    Some questions remain open
    
    
    
    THE PROBLEMS
      1    CAROUSEL NUMBERS (the appetizer problem)
      2    PERIODS OF DECIMALS (the main problem)
      3    NINE NUMBERS & ONE NUMBERS (useful aside)
      4    GROUP OF UNITS MOD N (final battleground)
    
    
    
    ------------------------------------------------------------------------
    1.     CAROUSEL NUMBERS  (the appetizer problem)
    ------------------------------------------------------------------------
    
    N = 142857 has a neat property: 
    
           142857                        142857
               x2                            x3
           ------                        ------
           285714                        428571
    
    We see the same digits and in the same order as the
    original 142857 but just cycled around. 
    
    Multiplying 142857 by 4, 5 and 6 gives the same result -
    the other three possible cyclic permutations of the 6
    digits. 
    
    Try 4N, 5N or 6N on your calculator. What is 7N?
    
    
    We call the number 142857 a CAROUSEL NUMBER.
    
    The next carousel number we know is
    
           M = 0588235294117647  
    
    It has 16 digits and we need the leading zero to make it
    work. Try multiplying it by 2 to see why.
    
           0588235294117647         0599235294117647
                         x2                      x
           ----------------         ----------------
           1176470588235294
    
    Here all the multiples from 2 through 16 produce cyclic
    permutations of the digits of M.
    
    Note this is too large a number for most calculators to
    handle. Do today's students still know how to multiply by
    hand? Do you? (Prove it by trying another case above.)
    
    Some natural questions arise:
    
           ARE THERE MORE?                                        (Yes, lots)
    
           HOW MANY?                                          (No one knows!)
    
           CAN ONE HAVE 2 DIGITS? ... 3 DIGITS?                 (No. Try it!)
    
           HOW DO WE FIND THEM?                                  (Read on...)
    
    
    --------------------------------------------------------------------------
    2.     PERIODS OF DECIMALS  (the real problem)
    --------------------------------------------------------------------------
    
    Most math students know the decimal for any rational
    number, m/n, will repeat. For example, 
    
    1/7 = .142857142857...              repeats with a period of 6, and
    
    5/12 = .41666...                    repeats with a period of 1 after a
                                        delay of 2. 
    1/13 = .076923...                   has period 6, half the max of 12
    
    1/17 = .05882352941887647...        has period 16 (the max it could have)
    
    
    RELATIONSHIP WITH CAROUSEL NUMBERS
    
    As hinted above with 1/7 and 1/17, every n where 1/n
    has maximal period n-1 will produce a carousel number.
    
    Only prime n's work and only some primes at that: 7, 17,
    19, 23, and 29 are the first five.
    
    We call these CAROUSEL PRIMES.
    
    
    Our main interest in the algebra/number theory course has been
    investigating what determines the period of decimals.
    
    This begins with long division done by hand, but quickly
    leads to writing computer programs to generate data and
    check out conjectures. Lots of questions yield to easy
    conjecture, but surprisingly no one knows the complete
    story. 
    
    It is too big a story to detail in this short talk, but here is
    one small part. We look at period of 1/p for prime p's.
    
           prime,p        p-1    per(1/p)    (p-1)/per(1/p)
           -------        ---    --------     ------------
              3            2       1              2
              7            6       6              1
             11           10       2              5
             13           12       6              2
             17           16      16              1
             19           18      18              1
             23           22      22              1
             29           28      28              1
             31           30      15              2
             37           36       3             12
    
    The conjecture that per(1/p) divides p-1 is later proved to
    be a consequence of Lagrange's theorem in group theory.
    (Which says the order of a subgroup divides the order of the group.)
    
    A similar "obvious" conjecture for per(1/p^a) turns out to
    be false - the first counterexample is for 1/487^2  (reference 1).
    
    
    USING THE COMPUTER
    
    Here is a program written in QBASIC to look for primes and then find the
    period of 1/p. It also shows (p-1)/per(1/p).
    
    'per(1/p).bas   a program to find the period of the decimal for 1/p
    '             (first find the primes, p, up to maxn)
    
    maxn = 289              'how far we will look for primes
    
    'first find the primes less than maxn by the sieve of Eritosthenes
    
    DIM n(maxn): n(1) = 0
    FOR j = 2 TO maxn: n(j) = 1: NEXT j: j = 1
    DO WHILE j * j < maxn
      j = j + 1
      IF n(j) = 1 THEN
        FOR k = 2 TO maxn / j: n(k * j) = 0: NEXT k
      END IF
    LOOP
    
    'now print out the primes and the period of 1/p
    
    numbase = 10: j = 2: m = 1
    PRINT "prime"; TAB(8); "per(1/p)"; TAB(18); "(p-1)/per"
    DO WHILE j < maxn
      IF n(j) > 1 AND numbase MOD j > 0 THEN
        remainder = 1
        place = 0
        DO
          place = place + 1
          remainder = numbase * remainder MOD j
        LOOP UNTIL remainder = 1
     PRINT TAB(2); j; TAB(10); place; TAB(18);
        FOR k = 1 TO (j - 1) / place: PRINT "*"; : NEXT k: PRINT
        m = m + 1
      END IF
    j = j + 1
    LOOP
    
    SOME QUESTIONS
            For which n's is the decimal for 1/n terminating?
                      (Find all the simple cases you can. What
                       determines the length of the decimal?)
            What determines when there is a delay? What determines
            the length of the delay?  (Relate to the question above.)
    
            How does per(1/p^a) relate to per(1/p)?
                       (Try small primes but 2 and 5 are special.)
            How does per(1/pq) relate to per(1/p), per(1/q)?
                       (Try 1/707. Note per(1/7)=6, per(1/101)=4.)
            How would you modify the program to just find per(1/n)?
    
    
    
    ----------------------------------------------------------------------------
    3.     NINE NUMBERS AND ONE NUMBERS  (a useful aside)
    ----------------------------------------------------------------------------
    
    The NINE NUMBERS, 9, 99, 999, 9999, etc., are important
    players in problem 2. Their prime factorizations show
    interesting patterns, and help explain the period of 1/n. Of
    course, every Nine number is nine times a ONE NUMBER,
    1, 11, 111, 1111, etc., so we concentrate on recognizing
    patterns in the prime factors of the One numbers. 
    
    We use Derive to help with factoring. (Derive is a nice,
    inexpensive, easy to learn computer algebra system.)
    
           99 = 911
           999 = 9337
           9999 = 911101
           99999 = 941271
           999999 = 937111337
    
    As a hint at the connection with periods of decimals,
           1/37 = 27/999 = .027027027... (period is 3).
    
    These factorizations help answer the question "What is the
    smallest n such that 1/n has period 4? ... has period 5?
    
    Note One numbers are also called "repunits". They have
    close connections with the polynomials
           1 + x + x^2 + ... + x^(n-1) = (x^n-1)/(x-1)
    In base 2 the prime One numbers are the Mersenne
    primes, famous for leading to perfect numbers.
    
    
    SAMPLE WORKSHEET
    
    Here is a sample worksheet we use in our course to help guide the
    discovery of patterns.
    
    Math 452    Worksheet on Factoring "One-numbers"                                      9/28/92
    
    I. Factoring the one-numbers: data generated by Derive
    
           n      prime factors of 111...1 (n ones)
          --      ---------------------------------
           2      11 (prime)
           3      3 37
           4      11 101
           5      41 271
           6      3 7 11 13 37
           7      239 4649
           8      11 73 101 137
           9      3 3 37 333667
           10     11 41 271 9091
           11     21649 5_____  (get into Derive and find the missing digits)
           12     3 7 11 13 37 101 9__1
           13     53 79 265371653
           14     11 239 4649 909091
           15     3 31 37 41 271 2906161
           16     11 17 73 101 137 5882353
           17     2071723 5363222357
           18     3 3 7 11 13 19 37 52579 333667
           19     1111111111111111111 (prime)
           20     11 41 101 271 3541 8091 27961
    
    II. Patterns and conjectures
           1.  11 divides a one-number for n =
           2.  3 and 37 divide for n =
           3.  101 divides for n =
           4.  41 and 271 divide for n =
           5.  What's really going on in 1-4 is that 
           6.  (Other conjectures?)
    
    MORE QUESTIONS
    
            Characterize all n's so that 1/n has period 1, period 2,
            period 3.
    
            What is the smallest n so 1/n has period 2, period 3, period 4?
            How is this related to factors of one numbers?
    
            Explain why Ones(k) = 11...1 (k ones) cannot be prime unless
            k is prime.
    
            Look up what the prime one numbers in base 2 have to do with
            perfect numbers.
    
    
    ----------------------------------------------------------------------------
    4.     THE GROUP OF UNITS MOD n, Un  (the final battleground)
    ----------------------------------------------------------------------------
    
    Un is the group of integers between 1 and n-1 that have
    no factors in common with n. It is a group under
    multiplication mod n. E.g. U7 = {1, 2, 3, 4, 5, 6} and
    2 x 5 = 10 = 3 (mod 7).
    
    When we do long division to find the decimal for 1/7
    (say), we are actually finding the powers of 10 in U7.
           
        .1428571...
       ------------
     7|1.0000000
         7
         30                                   10 = 3 (mod 7)
         28
          20                            100 = 30 = 2 (mod 7)
          14
           60                    1000 = 300 = 20 = 6 (mod 7)
           56
            40                 10,000 = ... = 60 = 4 (mod 7)
            35
             50                  10^5 = ... = 40 = 5 (mod 7)
             49
              10                 10^6 = ... = 50 = 1 (mod 7)
    
    That is, the remainder at every stage of the long division,
    3, 2, 6, 4, 5, 1, is the same as the successive powers of 10
    mod 7. AND THE LONG DIVISION PROCESS IS JUST REDUCTION OF POWERS
    OF 10 MOD 7 (or mod n in general).
    
    So per(1/n) = the order of 10 in Un and we may use all
    the power of group theory to attack the periods of
    decimals investigation. (The order of an element in a finite group
    is the smallest power which gives 1.)
    
    OTHER BASES
    
    Note that "decimals" in other bases are fairly difficult. For
    example, the "decimal" in base two for 1/7 involves writing 7 in
    base 2 as (111), and doing long division in base 2.
                 .001...
                -------
           111 |1.000...
                  111
                  ---
                    1
    Since we get the remainder of 1 here everything will repeat. That is,
    in base 2 the decimal for 1/7 = 1/(111) = .001001001... (base 2).
    
    However, finding the order of 2 in U7 is easy. 2^1 = 2, 2^2 = 4, and
    2^3 = 8 = 1 (mod 7). So we see "2 has order 3 in U7" exactly corresponds
    to "1/7 has period 3 as a binary decimal".
    
    EVEN MORE QUESTIONS
            Find the order of 10 in a bunch of Un's to check it is the
            same as per(1/n): pick n's with no 2's or 5's
    
            Explore decimals in other bases. Find a prime, p, which has
            period p-1 as a base 2 decimal. Check that the repeating
            block is indeed a base 2 carousel number
    
            Form a list of carousel primes and try to see what they
            have in common. Predict a new carousel prime and test it.
            (This is likely hard. Does that put you off, or inspire you?)
    
    
    ---------------------------------------------------------------------------
    5.     THE BIG CONNECTING THEOREM
    ---------------------------------------------------------------------------
    
    THEOREM
    Assume m/n is a reduced fraction and n' is the result of
    dividing all 2's and 5's out of n.
    
    Then the following are equivalent:
      i)   The decimal for m/n eventually repeats every k places
    
     ii)   The decimal for 1/n' repeats every k places (with no delay)
    
    iii)   1/n' can be written as B/99...9 (k 9's)
    
     iv)   n' divides 99...9 (k 9's)
    
      v)   10^k = 1 mod n'  That is, 10^k = 1 in Un'
    
    COROLLARY
    Thus the period of m/n (reduced) =
           per(1/n) =
           per(1/n') =
           the length of the smallest Nine number n' divides =
           the order of 10 in Un'
    
    APPLICATION TO CAROUSEL NUMBERS
    If p is a prime with 1/p having largest possible period p-1,
    then B = 99...9(p-1 9'S)/p is a carousel number.
     
    These carousel primes (primes where per(1/p)=p-1) are characterized by
      -    the first Nines number that p divides has length p-1
      -    10 is a generator of the cyclic group Up
    
    A better characterization is needed - we would like an
    easily checked property of p to make it a carousel prime.
    
    YET EVEN MORE PROBLEMS
            Prove these results
    
            Show that every carousel number arises in this way (as a
            Nines number over a carousel prime).
    
    
    ---------------------------------------------------------------------------
    PARTIAL CAROUSEL NUMBERS
    ---------------------------------------------------------------------------
    
    The number D = 076923 behaves a bit like a carousel number. It has 6
    digits (again we need the leading 0), and half of the multiples
    from 1D, 2D, 3D, ..., 12D are cyclic permutations of D.
    
    For example, 3D = 230769, and 4D = 307692 give the same digits in the same
    order. However 2D = 153846, and 5D = 384615 do not. (Note 5D is a cyclic
    permutation of 2D. Hmmmmm... check out the other multiples)
    
    So this number, D = 076923, is sort of "half" a carousel number. Not all
    the beginning multiples produce cyclic permutations, but half do.
    
    What is true about D = 076923 is that all of the cyclic permutations,
    769230, 692307, etc are multiples of D. (Check that.)
    
    DEFINITION
    So we might define a PARTIAL CAROUSEL NUMBER to be an integer (perhaps
    with leading 0's) with the property that all of its cyclic permutations
    are multiples of the number.
    
    CONNECTION WITH PERIODS OF DECIMALS
    Again there is a connection with periodic decimals. 1/13 = .076923...
    repeating with a period of 6 (half the max. potential period of 12).
    
    Looking at the powers of 10 in U13 (which are the same as the
    remainders in the long division process to find the decimal for 1/13)
    we get          10 = 10 (mod 13)
                    100 = 9 (mod 13) - subtract 7 13's = 91
                    10^3 = 90 = 12 (mod 13)
                    10^4 = 120 = 3 (mod 13)
                    10^5 = 30 = 4 (mod 13)
                    10^6 = 40 = 1 (mod 13).
    These are exactly the multiples that "work": 10D, 9D, 12D, 3D, 4D,
    (and 1D) give the 6 cyclic permutations of the digits of D.
    
    Further, they are in the order of how much cycling happens:
            10D = 769230  cycle or "left shift" by 1
             9D = 692307    left shift by 2 from D = 076923
            12D = 923076    left shift by 3
            etc.
    
    So we are on to something here. We can find more interesting numbers
    by not insisting that all the multiples 2X, 3X, ...dX be cyclic
    permutations, but only that all cyclic permutations be multiples.
    
    A PROBLEM ARISES
    Alas, the inclusion of leading 0's develops into a real problem.
    
    For example, 37 is not a partial carousel number because 73 is not
    a multiple of 37. But 037 is!
            (shift left 1) 370 = 10 times 37
            (shift left 2) 703 = 19 times 37
    
    It turns out that because 37 divides 999 (3 nines) we need the leading
    zero to have a 3 digit partial carousel number.
    
    THEOREM
    (nearly) EVERY NUMBER WILL BE A PARTIAL CAROUSEL NUMBER
    
    Idea of the proof: Nearly every number divides a Nines number. The
    length of the Nines number needed is related to the period of 1/n.
    Add enough leading 0's to match the length of that Nines number. You
    will now have a partial carousel number.
    
    EXAMPLE   Let n = 123. Factoring, n = 3 x 41. It turns out 1/3 has
            period 1 and 1/41 has period 5, so 1/123 has period 5. We
            claim 00123 has the partial carousel property. (We have added
            enough leading 0's to bring the length to 5.)
    
    COROLLARY
    Maybe partial carousel numbers aren't all that interesting after all.
    Oh well, you can't win them all.
    
    ---------------------------------------------------------------------------
    REFERENCES
    ---------------------------------------------------------------------------
    
    Carousel numbers and Carousel primes are my own names.
    
    So are Nine numbers and One numbers.
    
    The carousel stuff may be new, but many people rediscover the story of
    periods of decimals. That is so much more fun than looking it up.
    
    But if you would like to see more of the story, here are some references.
    
    1.      Solved and Unsolved Problems in Number Theory
            by Daniel Shanks, Chelsea Publ. Co., 2nd ed. 1978
    
    2.      Contemporary Abstract Algebra
            by Joseph A. Gallian, Heath, 2nd ed., 1990
    
    3.      A Concrete Introduction to Higher Algebra
            by Lindsay Childs, Springer-Verlag, 1979
    
    4.      Numbers: Rational and Irrational
            by Ivan Niven, MAA, 1961
    
    5.      The Theory of Numbers, An Introduction
            by Anthony A. Giola, Markham Publ. Co., 1970
    
    6.      Excursions into Mathematics
            by Beck, Bleicher, and Crowe, Worth Publ., 1969
    
    
    I have lots of fun with these investigations, and continue to do so.
    If you have any comments on the above, or ideas to add, I would
    REALLY like to hear from you.
    
    Dr. Gary Klatt
    Dept of Mathematics and Computer Science
    Univ of Wisconsin - Whitewater
    Whitewater, WI  53190
    email: klattg@uwwvax.uww.edu
    


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