## Orlando Presentation

### Carousel Numbers: A Lead-in to Number Theory

#### Gary Klatt

```------------------------------------------------------------------------
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 4łN, 5łN or 6łN on your calculator. What is 7łN?

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...)

```