The Girl who played with Fire

The extra batteries she had ordered for her Apple PowerBook (G4 titanium with a seventeen-inch screen) had finally arrived. In Miami she had bought a Palm PDA with a folding keyboard that she could use for email and easily take with her in her shoulder bag instead of dragging around her PowerBook, but it was a miserable substitute for the seventeen-inch screen. The original batteries had deteriorated and would run for only half an hour before they had to be recharged, which was a curse when she wanted to sit out on the terrace by the pool, and the electrical supply on Grenada left a lot to be desired. During the weeks she had been there, she had experienced two long blackouts. She paid with a credit card in the name of Wasp Enterprises, stuffed the batteries in her shoulder bag, and headed back out into the midday heat.

She paid a visit to Barclays Bank and withdrew $300, then went down to the market and bought a bunch of carrots, half a dozen mangoes, and a big bottle of mineral water. Her bag was much heavier now, and by the time she got back to the harbour she was hungry and thirsty. She considered the Nutmeg first, but the entrance to the restaurant was jammed with people already waiting. She went on to the quieter Turtleback at the other end of the harbour. There she sat on the veranda and ordered a plate of calamari and chips with a bottle of Carib, the local beer. She picked up a discarded copy of the Grenadian Voice and looked through it for two minutes. The only thing of interest was a dramatic article warning about the possible arrival of Matilda. The text was illustrated with a photograph showing a demolished house, a reminder of the devastation wrought by the last big hurricane to hit the island.

She folded the paper, took a swig from the bottle of Carib, and then she saw the man from room 32 come out on the veranda from the bar. He had his brown briefcase in one hand and a glass of Coca-Cola in the other. His eyes swept over her without recognition before he sat on a bench at the other end of the veranda and fixed his gaze on the water beyond.

He seemed utterly preoccupied and sat there motionless for seven minutes, Salander observed, before he raised his glass and took three deep swallows. Then he put down the glass and resumed staring out to sea. After a while she opened her bag and took out Dimensions in Mathematics.

All her life Salander had loved puzzles and riddles. When she was nine her mother gave her a Rubik’s Cube. It had put her abilities to the test for barely forty frustrating minutes before she understood how it worked. After that she never had any difficulty solving the puzzle. She had never missed the daily newspapers’ intelligence tests; five strangely shaped figures and the puzzle was how the sixth one should look. To her, the answer was always obvious.

In elementary school she had learned to add and subtract. Multiplication, division, and geometry were a natural extension. She could add up the bill in a restaurant, create an invoice, and calculate the path of an artillery shell fired at a certain speed and angle. That was easy. But before she read the article in Popular Science she had never been intrigued by mathematics or even thought about the fact that the multiplication table was math. It was something she memorized one afternoon at school, and she never understood why the teacher kept going on about it for the whole year.

Then, suddenly, she sensed the inexorable logic that must reside behind the reasoning and the formulas, and that led her to the mathematics section of the university bookshop. But it was not until she started on Dimensions in Mathematics that a whole new world opened to her. Mathematics was actually a logical puzzle with endless variations – riddles that could be solved. The trick was not to solve arithmetical problems. Five times five would always be twenty-five. The trick was to understand combinations of the various rules that made it possible to solve any mathematical problem whatsoever.

Dimensions in Mathematics was not strictly a textbook but rather a 1,200-page brick about the history of mathematics from the ancient Greeks to modern-day attempts to understand spherical astronomy. It was considered the bible of math, in a class with what the Arithmetica of Diophantus had meant (and still did mean) to serious mathematicians. When she opened Dimensions in Mathematics for the first time on the terrace of the hotel on Grand Anse Beach, she was enticed into an enchanted world of figures. This was a book written by an author who was both pedagogical and able to entertain the reader with anecdotes and astonishing problems. She could follow mathematics from Archimedes to today’s Jet Propulsion Laboratory in California. She had taken in the methods they used to solve problems.

Pythagoras’ equation (x² + y² = z²), formulated five centuries before Christ, was an epiphany. At that moment Salander understood the significance of what she had memorized in secondary school from some of the few classes she had attended. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. She was fascinated by Euclid’s discovery in about 300 BC that 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 she could see the endless combinations.

6 = 2¹x (2² ? l)

28 = 2²x (2³ ? l)

496 = 2⁴x (2⁵ ? l)

8,128 = 2⁶x (2⁷ ? l)

She could go on indefinitely without finding any number that would break the rule. This was a logic that appealed to her sense of the absolute. She advanced through Archimedes, Newton, Martin Gardner, and a dozen other classical mathematicians with unmitigated pleasure.

Then she came to the chapter on Pierre de Fermat, whose mathematical enigma, “Fermat’s Last Theorem,” had dumbfounded her for seven weeks. And that was a trifling length of time, considering that Fermat had driven mathematicians crazy for almost four hundred years before an Englishman named Andrew Wiles succeeded in unravelling the puzzle, as recently as 1993.

Fermat’s theorem was a beguilingly simple task.

Pierre de Fermat was born in 1601 in Beaumont-de-Lomagne in southwestern France. He was not even a mathematician; he was a civil servant who devoted himself to mathematics as a hobby. He was regarded as one of the most gifted self-taught mathematicians who ever lived. Like Salander, he enjoyed solving puzzles and riddles. He found it particularly amusing to tease other mathematicians by devising problems without supplying the solutions. The philosopher Descartes referred to Fermat by many derogatory epithets, and his English colleague John Wallis called him “that damned Frenchman.”

In 1621 a Latin translation was published of Diophantus’ Arithmetica which contained a complete compilation of the number theories that Pythagoras, Euclid, and other ancient mathematicians had formulated. It was when Fermat was studying Pythagoras’ equation that in a burst of pure genius he created his immortal problem. He formulated a variant of Pythagoras’ equation. Instead of (x² + y² = z²), Fermat converted the square to a cube, (x³ + y³ = z³).

The problem was that the new equation did not seem to have any solution with whole numbers. What Fermat had thus done, by an academic tweak, was to transform a formula which had an infinite number of perfect solutions into a blind alley that had no solution at all. His theorem was just that – Fermat claimed that nowhere in the infinite universe of numbers was there any whole number in which a cube could be expressed as the sum of two cubes, and that this was general for all numbers having a power of more than 2, that is, precisely Pythagoras’ equation.

Other mathematicians swiftly agreed that this was correct. Through trial and error they were able to confirm that they could not find a number that disproved Fermat’s theorem. The problem was simply that even if they counted until the end of time, they would never be able to test all existing numbers – they are infinite, after all – and consequently the mathematicians could not be 100 percent certain that the next number would not disprove Fermat’s theorem. Within mathematics, assertions must always be proven mathematically and expressed in a valid and scientifically correct formula. The mathematician must be able to stand on a podium and say the words This is so because…

Fermat, true to form, sorely tested his colleagues. In the margin of his copy of Arithmetica the genius penned the problem and concluded with the lines Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet. These lines became immortalized in the history of mathematics: I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.