|
| hosted by the Harvey Mudd College Math Department | created, authored and ©1999-2010 by Francis Su |
or
follow us on Twitter.
From the Fun Fact files, here is a Fun Fact at the Advanced level:Greedy to Avoid Progressions | ||||||||
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:
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:
Presentation Suggestions: The Math Behind the Fact: 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:
| ||||||||
Want another Math Fun Fact? For more fun, tour the Mathematics Department at Harvey Mudd College! 
| ||||||||
