A perfect number

After Christmas dinner with my husband's family a year ago I wrote a post about mathematical knitting, courtesy of retired mathematician Grandma Joan. That post is the most popular one I've ever written, according to my site meter, and it continues to get Google hits on a regular basis. Therefore I am sure the blogosphere will be delighted to know that I've returned from Christmas dinner with some more math for you. This time it comes courtesy of retired mathematician Cousin Ward.

Can you picture the scene? Imagine the cozy living room, fire crackling merrily, cousins playing happily at one end, Ward and I ensconced in a fabulous leather sofa at the other. "So," I say to him conversationally, "Tell me some more about perfect numbers." Being a mathematician, conversation is not his strong point and I have to do most of the work, though being an introverted bookworm, conversation is not my strong point either. But I am thinking that perfect numbers would make a great blog post, not only because they are interesting in themselves but also because the topic is a perfect way for me to casually let slip that my birthday is tomorrow -- on the TWENTY-EIGHTH of December.

In fact, I first learned about perfect numbers at a long-ago family dinner when I confessed to Ward my life-long fascination with the number 28. He could not have made me happier than when he told me that mathematically speaking 28 is a rare bird -- a "perfect" number. Six is a perfect number too. And the next one after 28 is four-hundred-and-something.

A perfect number is one that equals the sum of its factors. Factors of 6 = 1, 2, 3. Factors of 28 = 1, 2, 4, 7, 14. Factors of four-hundred-and-something? Um. This would make a much better post if I knew four-hundred-and-what, which is why I ask Ward: "Four-hundred-and-what?" Suddenly he is staring off into space, lips moving slightly. I stare at his skull, wondering what's going on inside.

"Well," he says finally, "What's 31 x 16? Whatever is 31 x 16, that's a perfect number." By this time Uncle Brad is listening too, and the three of us attempt to multiply 31 x 16 in our heads. Of course Ward comes up with the answer first: 496. There is some joking about whether he'd get the same answer in subsequent attempts, but -- check it yourself -- he was right.

Meanwhile, I am astounded. What does 31 x 16 have to do with anything? I press Ward for details and he tells me that there's a formula for finding perfect numbers. He tells me the formula but I have trouble hearing because Daniel is busily drilling my knee with a fairly loud toy electric drill. But this is too good to let go. "Hold that thought," I say to Ward. I push Daniel away, and run off to find my dear hubby. And of course he has pen and paper on hand, because he is Steve. So I run back to Ward with pen and paper, and here it is:

A perfect number equals (2^p-1)(2^p-1), where p is a prime number.

So if p=3 you get 2³ = 8, minus 1 is 7, and 2² = 4. And voilà! 7 x 4 = 28! And when p = 5 you get 31 x 16.

I ask Ward whether perfect numbers are, you know, useful. Do they have any practical application?

Ward laughs. "None whatsoever."