This is the summary of a presentation given at the Joint Mathematics
Meetings, January 10-13, 1996, Orlando, Florida.
Carousel Numbers - A Lead-in to Number Theory
We examine a connected series of four problems in elementary number theory that
are ideal for discovery learning at several levels. Each problem generates
interesting questions and conjectures, and their surprising connections add
interest and allow each to shed light on the others. That some questions are
open adds challenge.
- Carousel Numbers
The number N = 142857 has a neat property: when you
it by 2 through 6 you get the same digits in the same order but cycled around.
5N = 714285 for example. The next such "carousel number" we know is
0588235294117647. Multiply it by 2 to see why we need the leading 0. We
to generate lots more, but no one knows if there are an infinite number of them.
- Periods of Decimals
That decimals for rational numbers are periodic is well
known. It is surprising that we know little more than Gauss did about what
determines these periods. The decimal for 1/7 is 0.142857142857... and has
period 6. That 1/7 has maximal period and 1/11 = 0.090909... doesn't hints at
the connection with carousel numbers.
- One Numbers
They are 11, 111, 1111, etc. Their prime factorizations have
patterns. They illuminate problem 2.
- The Group of Units mod n
We use the power of group theory to get more
and prove some of our conjectures.
Dr. Gary Klatt
Dept of Mathematics and Computer Science
Univ of Wisconsin - Whitewater
Whitewater, WI 53190