An arithmetic progression is a sequence of
3 or more integers whose terms differ by a constant, e.g.,
20, 23, 26, 29 is an arithmetic progression.
Question:
does every increasing sequence of integers
have to contain an arithmetic progression in it?
For instance, the sequence
of primes 2, 3, 5, 7, 11, 13,...
contains an arithmetic progression: 3,7,11.
Somewhat surprisingly, there are sequences with
no progressions! We'll construct an example:
Start with 0. Then for the next term in the
sequence, be greedy:
take the smallest possible integer
that does not cause an arithmetic progression to form
in the sequence constructed so far. (There must
be such an integer because there are infinitely
many integers beyond the last term, and only finitely
many possible progressions the new term could complete.)
This gives:
0, 1, 3, 4, 9, 10, 12, 13, 27, 28,...
Clearly this sequence has the required property.
Now that we have a solution, is there a better way
to understand what this sequence is,
without having to rely on recursion?
Yes. The above sequence can also be obtained by
writing all the positive integers in base 2, then
interpreting them in base 3! The base 2 integers are:
0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001,...
and if you think of these as base 3 numbers, you get
the above integers!
Presentation Suggestions:
Encourage students to think about why the binary trick
works.
The Math Behind the Fact:
There are two important mathematical ideas here.
The trick used to construct the above sequence is
an example of a greedy algorithm.
Greedy algorithms arise as solutions to
many problems in computer science
and mathematics.
The second lesson is that often when a solution
is developed, we can find a simpler one by insight:
it is a nice exercise to show that
the binary trick work because in base 3,
if any two terms contain just 0's and 1's, then
a third term that completes an arithmetic progression
must contain a 2!
How to Cite this Page:
Su, Francis E., et al. "Greedy to Avoid Progressions."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.

