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3.     NINE NUMBERS AND ONE NUMBERS  (a useful aside)
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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 = 9ù11
       999 = 9ù3ù37
       9999 = 9ù11ù101
       99999 = 9ù41ù271
       999999 = 9ù3ù7ù11ù13ù37

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.

Go on to the next section