Her true identity only became clear to Gauss when Napoleon’s French army invaded Prussia in 1806. Germain was anxious that Gauss, like Archimedes, might become the victim of a military invasion, so she sent a message to General Joseph-Marie Pernety, a family friend who was commanding the advancing forces. He duly guaranteed Gauss’s safety, and explained to the mathematician that he owed his life to Mademoiselle Germain. When Gauss realized that Germain and LeBlanc were the same person, he wrote:
But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur LeBlanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it: the enchanting charms of this sublime science reveal themselves only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents, and superior genius.
In terms of pure mathematics, Germain’s most famous contribution was in relation to Fermat’s last theorem. Although she could not formulate a complete proof, Germain made more progress than anyone else of her generation, which prompted the Institut de France to award her a medal for her achievements.
She also had an interest in prime numbers, those numbers that cannot be divided by any other number except 1 and the number itself. Prime numbers can be put into different categories, and one particular set is named in honor of Germain. A prime number p is labeled a Germain prime if 2p + 1 is also prime. So, 7 is not a Germain prime, because 2 × 7 + 1 = 15, and 15 is not prime. By contrast, 11 is a Germain prime, because 2 × 11 + 1 = 23, and 23 is a prime.
Research into prime numbers is nearly always considered important, because these numbers are essentially the building blocks of mathematics. In the same way that all molecules are composed of atoms, all the counting numbers are either primes or the products of primes multiplied together. Given that they are central to all things numerical, it will not come as a surprise that a prime number makes a guest appearance in a 2006 episode of The Simpsons, as we will discover in the next chapter.
Examination II
HIGH SCHOOL PAPER
Joke 1
Q: What are the 10 kinds of people in the world?
1 point
A: Those who understand binary, and those who don’t.
Joke 2
Q: Which trigonometric functions do farmers like?
1 point
A: Swine and cowswine.
Joke 3
Q: Prove that every horse has an infinite number of legs.
2 points
A: Proof by intimidation: Horses have an even number of legs. Behind they have two legs and in front they have forelegs. This makes a total of six legs, but this is an odd number of legs for a horse. The only number that is both odd and even is infinity. Therefore horses have an infinite number of legs.
Joke 4
Q: How did the mathematician reply when he was asked how his pet parrot died?
2 points
A: Polynomial. Polygon.
Joke 5
Q: What do you get when you cross an elephant and a banana?
3 points
A: | elephant | × | banana | × sin θ
Joke 6
Q: What do you get if you cross a mosquito with a mountain climber?
3 points
A: You can’t cross a vector with a scalar.
Joke 7
One day, Jesus said to his disciples: “The Kingdom of Heaven is like 2x2 + 5x – 6.” Thomas looked confused and asked Peter: “What does the teacher mean?”
Peter replied: “Don’t worry— it’s just another one of his parabolas.”
2 points
Joke 8
Q: What is the volume of a pizza of thickness a and radius z?
3 points
A: pi.z.z.a
Joke 9
During a security briefing at the White House, Defense Secretary Donald Rumsfeld breaks some tragic news: “Mr President, three Brazilian soldiers were killed yesterday while supporting U.S. troops.”
“My God!” shrieks President George W. Bush, and he buries his head in his hands. He remains stunned and silent for a full minute. Eventually, he looks up, takes a deep breath, and asks Rumsfeld: “How many is a brazillion?”
3 points
TOTAL - 20 POINTS
CHAPTER 8
A Prime-Time Show
The storyline of “Marge and Homer Turn a Couple Play” (2006) centers around a baseball star named Buck “Home Run King” Mitchell, who plays for the Springfield Isotopes. When he and his wife, Tabitha Vixx, experience marital problems, Mitchell’s performance on the field begins to suffer, so they turn to Homer and Marge for relationship advice. After various twists and turns, the episode culminates at Springfield Stadium, where Tabitha hijacks the Jumbo-Vision screen and publicly declares her love for Buck to the entire crowd.
The episode features the voice of singer and actress Mandy Moore, a reference to J. D. Salinger, and a nod to Michelangelo’s Pietà, but mathematical viewers would have been most excited by an appearance by a very special prime number. Before revealing the details of the prime number and how it is incorporated into the episode, let us step back and meet the two mathematicians who provided the inspiration for this prime number reference, namely Professor Sarah Greenwald of Appalachian State University and Professor Andrew Nestler of Santa Monica College.
Greenwald and Nestler’s interest in The Simpsons dates back to 1991, when they first met and became friends at the Mathematics Department at the University of Pennsylvania. They were both starting work on their PhDs, and once a week they would gather with other graduate students to watch The Simpsons and share a meal. Nestler remains clear about why the series appealed to them: “The writers created two recurring nerds: Professor Frink, a scientist, and Martin Prince, a gifted elementary school student. And they were alongside a main character, Lisa Simpson, who was also highly intelligent and inquisitive. The inclusion of these characters made the show something that intellectuals would want to watch in order to, in a sense, laugh at themselves.”
It was not long before Greenwald and Nestler began to pick up on the various mathematical references in The Simpsons. As well as enjoying the jokes about higher mathematics, they were tickled by those scenes involving mathematics in the context of education. Nestler recalls that he was particularly fond of a line by Edna Krabappel, in “This Little Wiggy” (1998), when Springfield’s bitterest teacher turns to her class and asks: “Now, whose calculator can tell me what seven times eight is?”
After a while, they encountered so many mathematical jokes that Nestler decided to create a database of scenes that might interest mathematicians. According to Nestler, it was the obvious thing to do: “I am by nature a collector, and enjoy cataloging things. When I was young I collected business cards. My main hobby is collecting Madonna records; I have over 2,300 physical records in my Madonna collection.”
A few years later, after they had received their doctorates and started teaching, both Greenwald and Nestler began incorporating scenes from The Simpsons into their lectures. Nestler, whose doctoral thesis was on algebraic number theory, used material from the animated sitcom in his courses covering calculus, precalculus, linear algebra, and finite mathematics.
By contrast, Greenwald’s research interest has always been orbifolds, a specialty within geometry, so she tended to include geometrical jokes from The Simpsons in her course titled Math 1010 (Liberal Arts Math). For example, she has discussed the opening couch gag from “Homer the Great” (1995). The opening sequence of each episode ends with the Simpso
n family converging on their couch in order to watch television, which always leads to a piece of visual humor. In this case, the couch gag involves Homer and his family exploring a paradoxical network of staircases under the influence of three gravitational forces, each one acting perpendicular to the others. This scene is a tribute to Relativity, a famous lithograph print by the twentieth-century Dutch artist M. C. Escher, who was obsessed with mathematics in general and geometry in particular.
After a few years of incorporating The Simpsons into their mathematics courses, Greenwald and Nestler’s quirky approach to teaching attracted some local media attention, which then led to an interview on National Public Radio’s Science Friday. When some of the Simpsons writers heard the show, they were astonished to learn that their nerdy inside jokes were now the basis of college mathematics courses. They were keen to meet the professors and thank them for their dedication to both mathematics and The Simpsons, so the writers invited Greenwald and Nestler to attend a table-read of an upcoming episode, which turned out to be “Marge and Homer Turn a Couple Play.”
On August 25, 2005, Greenwald and Nestler listened to the table-read that described the topsy-turvy relationship between Buck Mitchell and Tabitha Vixx. While the professors sat back and enjoyed the story, the writers paid close attention to every line, listening for good gags that could be made better and bad gags that ought to be dropped. Later that day, after the professors had returned home, the writers compared notes and began to offer tweaks to the script. Everyone around the table agreed that this was a strong episode, but there was one glaring omission—the entire episode was devoid of mathematics!
It seemed rude to have invited Greenwald and Nestler to a table-read because of their interest in the mathematics of The Simpsons, yet show the professors an episode that would not provide them with any new material for their classes. The writers started re-examining the script, scene by scene, looking for an appropriate place to insert some mathematics. Eventually, one of them spotted that the climax of the episode provided the perfect opportunity to bring in some interesting numbers.
Just before Tabitha makes her declaration of love on the Jumbo-Vision screen, a question is displayed on the same screen that asks the crowd to guess the attendance at the game. It is presented as a multiple-choice question. In the table-read script, the numbers on offer on the screen were just plucked out of the air, but now the writers set about replacing them with numbers that possessed particularly interesting properties. Once the writers had completed their mission, Jeff Westbrook e-mailed Sarah Greenwald: “It’s great you guys came by, because it really did light a little fire under us to some degree and today we put in some slightly more interesting mathematical numbers in honor of your visit.”
The layout of the Jumbo-Vision screen from “Marge and Homer Turn a Couple Play.”
The three interesting numbers, as they appeared on the Jumbo-Vision screen, would have seemed arbitrary and innocuous to casual viewers, but those with mathematical minds would immediately have seen that each one is remarkable in its own way.
The first number, 8,191, is a prime number. Indeed, it belongs to a special class of prime numbers known as Mersenne primes. These are named after Marin Mersenne, who joined the Minim friars in Paris in 1611, thereafter dividing his time between praying to God and worshipping mathematics. He became particularly interested in a set of numbers of the form 2p – 1, where p is any prime number. The table below shows what happens if you plug all the prime numbers less than 20 into the formula 2p – 1.
Prime (p)
2p – 1
Prime?
2
22 – 1 =
3
✓
3
23 – 1 =
7
✓
5
25 – 1 =
31
✓
7
27 – 1 =
127
✓
11
211 – 1 =
2,047
✘
13
213 – 1 =
8,191
✓
17
217 – 1 =
131,071
✓
19
219 – 1 =
524,287
✓
The striking feature in the table is that 2p – 1 seems to generate prime suspects, by which I mean numbers that might be prime. Indeed, all the numbers in the right-hand column are primes, except 2,047, because 2,047 = 23 × 89. In other words, 2p – 1 is a recipe that uses prime numbers as its ingredients in an attempt to make new prime numbers; these resulting primes are dubbed Mersenne primes. For example, when p = 13, then 213 – 1 = 8,191, which is the Mersenne prime that appears in “Marge and Homer Turn a Couple Play.”
Mersenne primes are considered celebrities within the world of numbers, because they can be very large. Some are titanic primes (more than one thousand digits), some are gigantic primes (more than ten thousand digits), and the very largest are labeled megaprimes (more than one million digits). The ten largest known Mersenne primes are the biggest primes ever identified. The largest Mersenne prime (257,885,161 – 1), which was discovered in January 2013, is more than seventeen million digits long.13
The second number on the stadium screen is 8,128, which is known as a perfect number. Perfection in the context of a number depends on its divisors, namely those numbers that will divide into it without any remainder. For example, the divisors of 10 are 1, 2, 5, and 10. A number is considered perfect if its divisors (except the number itself) add up to the number in question. The smallest perfect number is 6, because 1, 2, and 3 are divisors of 6, and 1 + 2 + 3 = 6. The second perfect number is 28, because 1, 2, 4, 7, and 14 are divisors of 28, and 1 + 2 + 4 + 7 + 14 = 28. The third perfect number is 496, and the fourth perfect number is 8,128, which is the one that crops up in “Marge and Homer Turn a Couple Play.”
These four perfect numbers were all known to the ancient Greeks, but mathematicians would have to wait more than a millennium before the next three perfect numbers were discovered. 33,550,336 was discovered in roughly 1460, then 8,589,869,056 and 137,438,691,328 were both announced in 1588. As René Descartes, the seventeenth-century French mathematician, pointed out, “Perfect numbers, like perfect men, are very rare.”
Because they are few and far between, it is easy to jump to the conclusion that there are only a finite number of perfect numbers. However, as yet, mathematicians cannot prove that the supply of perfect numbers is limited. Also, all the perfect numbers discovered so far are even, so perhaps all future perfect numbers will be even. Again, as yet, nobody has proved that this is indeed the case.
Despite these holes in our knowledge, we do know a few things about perfect numbers. For example, it has been proved that perfect numbers that are even (which might be all of them) are also triangular numbers:
6 = 1 + 2 + 3
28 = 1 + 2 + 3 + 4 + 5 + 6 + 7
Moreover, we know that even perfect numbers (except 6) are always the sum of a series of consecutive odd cubes:
28
=
13 + 33
496
=
13 + 33 + 53 + 73
8,128
=
13 + 33 + 53 + 73 + 93 + 113 + 133 + 153
Last, but certainly not least, we know that there is a close relationship between even perfect numbers and Mersenne primes. In fact, mathematicians have proved that there is the same number of each, and it has been shown that every Mersenne prime can be used to generate a perfect number. Hence, we know of only forty-eight perfect numbers, because we know of only forty-eight Mersenne primes.
The third number that appears on the stadium screen, 8,208, is special because it is a so-called narcissistic number. This means that the number is equal to the sum of each of its digits raised to the power of the number of digits:
8,208 = 84 + 24 + 04 + 84 = 4,096 + 16 + 0 + 4,096
The reason why
this number is labeled narcissistic is that the digits within it are being used to generate the number itself. The number appears to be self-obsessed, almost in love with itself.
There are many other examples of narcissistic numbers, such as 153, which equals 13 + 53 + 33, but it has been shown that there is only a finite supply of narcissistic numbers. In fact, mathematicians know that there are only eighty-eight narcissistic numbers, and the largest one is 115,132,219,018,763,992,565,095,597,973,971,522,401.
However, if we relax the constraints, then it is possible to generate so-called pretty wild narcissistic numbers; these numbers can be generated using their own digits in any way that works. Here are some examples of pretty wild narcissistic numbers:
6,859
=
(6 + 8 + 5)√9
24,739
=
24 + 7! + 39
23,328
=
2 × 33! × 2 × 8