## The Simpsons' secret formula: it's written by maths geeks | The Observer

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Simpsons and math geeks

The screen displays three multiple choice options; 8,128, 8,208 and 8,191. These digits might seem arbitrary and innocuous, but in fact they represent a perfect number, a narcissistic number and a Mersenne prime.

8,128 is called a perfect number, because its divisors add up to the number itself. The smallest perfect number is 6, because 1, 2 and 3 not only divide into 6, but they also add up to 6. The second perfect number is 28, because 1, 2, 4, 7 and 14 not only divide into 28, but they also add up to 28. The third perfect number is 496, and the fourth one is 8,128, which appears in this episode. As René Descartes, the 17th-century French mathematician (and philosopher) pointed out: "Perfect numbers, like perfect men, are very rare."

8,208 is a narcissistic number because it contains 4 digits, and raising each of these digits to the 4th power generates four numbers that add up to itself: 84 + 24 + 04 + 84 = 8,208.

The fact that 8,208 can recreate itself from its own components hints that the number is in love with itself, hence the narcissistic label. Among the infinity of numbers, fewer than 100 exhibit narcissism.

8,191 is a prime number, because it has no divisors other than 1 and the number itself, and it is labelled a Mersenne prime because another 17th-century French mathematician, Marin Mersenne, spotted that 8,191 was equal to 213 – 1. More generally, Mersenne primes fit the pattern 2p –1, where p is any prime number.

Not surprisingly, several of the mathematical quips in The Simpsons relate to Homer and Marge's daughter, Lisa. She is proud to be a nerd and her grasp of everything from trigonometry to logarithms is recognised by Principal Skinner in the episode "Treehouse of Horror X" (1999). After a stack of bench seats falls on Lisa, he cries out: "She's been crushed! And so have the hopes of our mathletics team."

In "MoneyBart", much of the episode is dedicated to Lisa's ruthlessly mathematical approach to coaching a winning baseball team. The entire storyline is rooted in statistics, but the most significant nerd reference appears and disappears in a blink of an eye. Just before her first big game in charge, we see Lisa poring over piles of technical books. This extraordinary sight prompts a reporter to remark: "I haven't seen this many books in a dugout since Albert Einstein went canoeing." Those who concentrate hard enough will spot that one of the books is titled "e^iπ + 1 = 0". To the untrained eye, this is just another random equation. To the mathematical eye, this is the single most beautiful equation in history, because it combines five of the fundamental ingredients of mathematics (0, 1, e, i, and π) in one elegant recipe. It is known as Euler's identity, and named after the 18th-century Swiss genius Leonhard Euler.

What e^pi*i means: http://www.math.toronto.edu/mathnet/questionCorner/epii.html

Edward Kasner and James Newman in Mathematics and the Imagination note, "We can only reproduce the equation and not stop to inquire into its implications. It appeals equally to the mystic, the scientists, the mathematician." This formula of Leonhard Euler (1707-1783) unites the five most important symbols of mathematics: 1, 0, pi, e and i (the square root of minus one). This union was regarded as mystic union containing representatives from each branch of the mathematical tree: arithmetic is represented by 0 and 1, algebra by the symbol i, geometry by pi, and analysis by the transcendental e. Harvard mathematician Benjamin Pierce said about the formula, "That is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."