## Archive for August 26, 2010

### The Girl With the Dragon Tattoo

*The Girl Who Played with Fire* is the second volume in the late Stieg Larsson‘s *The Millenium Trilogy*. (Of course, you probably already knew that, since virtually everyone in North America has read this book. I mean, *someone* had to buy those 20 million copies, right?)

In this book the heroine, Lisbeth Salander, gets absorbed in recreational mathematics. She stumbles across a theorem about perfect numbers that, surprisingly, was proved by Euclid. (This is surprising because Euclid did most of his work in geometry, and a proof of his theorem about perfect numbers would rely on algebra and number theory.) The theorem appeared as Proposition IX.36 of Euclid’s *Elements.*

Stieg Larsson writes:

…a perfect number is always a multiple of two numbers, in which one number is a power of 2, and the second consists of the difference between the next power of 2 and 1. This was a refinement of Pythagoras’ equation, and [Lisbeth] could see the endless combinations:

6 = 2

^{1}(2^{2}– 1)28 = 2

^{2}(2^{3}– 1)496 = 2

^{4}(2^{5}– 1)8128 = 2

^{6}(2^{7}– 1)She could go on indefinitely without finding any numbers that would break the rule.

What Lisbeth does not state, but what is required for Euclid’s theorem to hold, is that 2^{k}(2^{k – 1} – 1) is a perfect number if and only if 2^{k – 1} is prime. She doesn’t state this — but her list of “endless combinations” only includes examples for which this is the case.

I don’t begrudge Larsson for this omission. After all, how can you be mad at the first author to sell more than one-million e-books on Amazon, especially when his most popular works were published posthumously? Besides, adding too much math to a popular fiction novel might make it a little less popular. I’m just happy that so many readers will be exposed to a little of the mathematical beauty that makes me love numbers.

Here’s a perfect quote from Descartes:

Perfect numbers, like perfect individuals, are very rare.

And a perfect joke:

Teacher: What is 14 + 14?

Student: 28.

Teacher: That’s good!

Student: Good? It’sperfect!