----------------------------------------------------------------------------
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 = 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.