Girl Who Played with Fire, The - Stieg Larsson [9]
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