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 = 21(22 – 1)
28 = 22(23 – 1)
496 = 24(25 – 1)
8128 = 26(27 – 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 2k(2k – 1 – 1) is a perfect number if and only if 2k – 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?
Teacher: That’s good!
Student: Good? It’s perfect!