Fun with abundant numbers

A few weeks ago Tracy Johnston Zager asked about this problem on twitter (and these embedded tweets should be amusing during Halloween week!):

Her post actually played a role in two blog entries so far:

A Neat number theory problem for kids from Tracy Johnston Zager


A neat number theory problem from David Radcliffe

Today’s post was inspired by one of the responses to her question which sort of stuck in the back of my mind:

That response moved from the back of my mind to the front when when my younger son and I got to section 6.4 of Art of Problem Solving’s Introduction to Number Theory book:  Perfect, Abundant, and Deficient Numbers.  Ha – maybe 10 days apart, but a funny coincidence nonetheless.  Time to see if there is indeed fun to be had with these numbers.

We ended up spending two days in this section because there was so much to cover.  As I’ve written a few times recently, finding ways to build up number sense has been on my mind and playing around with factoring (and then adding up the factors) seems like a perfectly fine way to spend time playing around with numbers.   So building number sense was definitely of the goals here.  One other thing that I thought would be fun in this section was talking about why people are interested in these properties of numbers in the first place.

As an aside, I was happy to see this post from Justin Lanier on twitter the other day discussing a time when he and his class worked through one amazingly complicated equation from a theoretical math paper about the Goldbach conjecture:

I think it is fun to try to figure out ways for kids see examples from current math research and I am going to try to put together a special project about prime numbers this weekend after seeing Justin’s post.

Anyway, back to abundant numbers.  To show my son one example where abundant numbers appear in number theory we looked at the Wikipedia page about super abundant number:

Superabundant Numbers

The first equation on that page looks pretty intimidating, but explaining that equation turned out to be a great way to talk about abundant and super abundant numbers with a kid.  Of course, in the background we got lots of good practice with numbers – finding factors, dividing, fractions, and sums.  Yes!!:

Since my son really seemed to enjoy talking about super abundant numbers the next day we went through a similar computation to see if we could find all of the numbers from 1 to 20 that were abundant.  I’m not aware of any really quick way to do this exercise other than checking most of the numbers individually (as my son points out in the video, you don’t have to check the primes).  Although this might be a dull exercise in isolation, he seems to be pretty excited about finding the abundant numbers and he doesn’t appear to find all the arithmetic work here to be dull.

Conclusion:  Twitter was right –  there was indeed some fun to be had with abundant numbers.  Of course, no one would think that a deep dive into the number theory would be appropriate for elementary school kids, but a deep dive isn’t necessary.  As I said, there’s a nice opportunity for building up number sense here and even a lucky opportunity to mention some advanced math.  All in all a neat topic and a fun couple of days!