The Number Mysteries: A Mathematical Odyssey through Everyday Life
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But what has all this rice got to do with finding big prime numbers? Ever since the Greeks had proved that the primes go on forever, mathematicians had been on the lookout for clever formulas that might generate bigger and bigger primes. One of the best of these formulas was discovered by a French monk named Marin Mersenne. Mersenne was a close friend of Pierre de Fermat and René Descartes, and he functioned like a seventeenth-century Internet hub, receiving letters from scientists all across Europe and communicating ideas to those he thought could develop them further.
His correspondence with Fermat led to the discovery of a powerful formula for finding huge primes. The secret of this formula is hidden inside the story of the rice and the chessboard. As you count up the grains of rice from the first square of the chessboard, the cumulative total quite often turns out to be a prime number. For example, after three squares, there are 1 + 2 + 4 = 7 grains of rice—a prime number. By the fifth square, there are 1 + 2 + 4 + 8 + 16 = 31 grains of rice.
Mersenne wondered whether it would turn out to be true that whenever you landed on a prime-number square on the chessboard, the number of grains of rice up to that point might also be a prime. If it were, it would give you a way of generating bigger and bigger primes. Once you’d counted the prime-number grains of rice, you’d just move to this square on the chessboard and count the number of grains of rice up to this point, which Mersenne hoped would be an even bigger prime.
Unfortunately for Mersenne and for mathematics, his idea didn’t quite work. When you look at the eleventh square of the chessboard, a prime-number square, then up to that point there are a total of 2,047 grains of rice. Sadly, 2,047 is not prime—it equals 23 × 89. But although Mersenne’s idea didn’t always work, it has led to some of the largest prime numbers that have been discovered.
THE GUINNESS BOOK OF PRIMES
During the reign of Queen Elizabeth I, the largest known prime number was the number of grains of rice on the chessboard up to and including the nineteenth square: 524,287. By the time Lord Nelson was fighting the Battle of Trafalgar, the record for the largest prime had gone up to the thirty-first square of the chessboard: 2,147,483,647. This ten-digit number was proved to be prime in 1772 by the Swiss mathematician Leonhard Euler, and it held the record until 1867.
By September 4, 2006, the record had gone up to the number of grains of rice that would be on the board up to square number 32,582,657, if we had a big enough chessboard. This new prime number has over 9.8 million digits, and it would take a month and a half to read it out loud. It was discovered not by some huge supercomputer but by an amateur mathematician using some software downloaded from the Internet.
The idea of this software is to utilize a computer’s idle time to do computations. The program it uses implements a clever strategy that was developed to test whether Mersenne’s numbers are prime. It still took a desktop computer several months to check Mersenne numbers with 9.8 million digits, but this is a lot faster than methods for testing whether a random number of this size is prime. By 2009, over ten thousand people had joined what has become known as the Great Internet Mersenne Prime Search, or GIMPS.
If you want your computer to join the GIMPS, you can download the software at www.mersenne.org , or scan this code with your smartphone.
Be warned, though, that the search is not without its dangers. One GIMPS recruit worked for a US telephone company and decided to enlist the help of 2,585 of the company’s computers in his search for Mersenne primes. The company began to get suspicious when its computers were taking five minutes rather than five seconds to retrieve telephone numbers. When the FBI eventually found the source of the slowdown, the employee owned up: “All that computational power was just too tempting for me,” he admitted. The telephone company didn’t feel sympathetic to the pursuit of science, and fired the employee.
After September 2006, mathematicians were holding their breath to see when the record would pass the ten million–digit barrier. The anticipation was not just for academic reasons—a prize of $100,000 was waiting for the person who got there first. The prize money was put up by the Electronic Frontier Foundation, a California-based organization that encourages collaboration and cooperation in cyberspace.
It was two more years before the record was broken. In a cruel twist of fate, two record-breaking primes were found within a few days of each other. The German amateur prime-number sleuth Hans-Michael Elvenich must have thought he’d hit the jackpot when his computer announced on September 6, 2008, that it had just found a new Mersenne prime with 11,185,272 digits. But when he submitted his discovery to the authorities, his excitement turned to despair—he had been beaten to it by 14 days. On August 23, Edson Smith’s computer in the math department at the University of California at Los Angeles (UCLA) had discovered an even bigger prime—one with 12,978,189 digits. For UCLA, breaking prime-number records was nothing new. UCLA mathematician Raphael Robinson discovered five Mersenne primes in the 1950s, and two more were found by Alexander Hurwitz at the beginning of the 1960s.
The developers of the program used by GIMPS agreed that the prize money shouldn’t simply go to the lucky person who was assigned that Mersenne number to check. Instead, $5,000 went to the developers of the software, $20,000 was shared among those who had broken records with the software since 1999, $25,000 was given to charity, and the rest went to Edson Smith in California.
If you still want to win money by looking for primes, don’t worry about the fact that the ten million–digit mark has been passed. For each new Mersenne prime found, there is a prize of $3,000. But if it’s the big money you’re after, there is $150,000 on offer for passing one hundred million digits and $200,000 if you can make it to the billion-digit mark. Thanks to the ancient Greeks, we know that such record primes are waiting out there for someone to discover them. It’s just a matter of how much inflation will have eaten into the worth of these prizes before someone eventually claims the next one.
How to Write a Number with 12,978,189 Digits
Edson Smith’s prime is phenomenally large. It would take over three thousand pages of this book to record its digits, but luckily, a bit of mathematics can produce a formula that expresses the number in a much more succinct manner.
The total number of grains of rice up to the Nth square of the chessboard is
R = 1 + 2 + 4 + 8 + . . . + 2N – 2 + 2N – 1.
Here’s a trick to find a formula for this number. It looks totally useless at first glance because it is so obvious: R = 2 R – R. How on earth can such an obvious equation help in calculating R? In mathematics, it often helps to take a slightly different perspective, because then everything can suddenly look completely different.
Let’s first calculate 2 R. That just means doubling all the terms in the big sum. But the point is that if you double the number of grains of rice on one square, the result is the same as the number of grains on the next square along. So
2R = 2 + 4 + 8 + 16 + . . . + 2N – 1 + 2N
The next move is to subtract R. This will just knock out all the terms of 2 R except the last one:
So the total number of grains of rice up to the Nth square of the chessboard is 2N – 1, and this is the formula responsible for today’s record-breaking primes. By doubling enough times and then taking 1 away from the answer, you might just hope to hit a Mersenne prime, as primes found using this formula are called.
HOW TO CROSS THE UNIVERSE WITH A DRAGON NOODLE
Rice is not the only food associated with exploiting the power of doubling to create large numbers. Dragon noodles, or la mian noodles, are traditionally made by stretching the dough between your arms and then folding it back again to double the length. Each time you stretch the dough, the noodle becomes longer and thinner, but you need to work quickly because the dough dries out quickly, disintegrating into a noodly mess.
Cooks across Asia have competed for the accolade of doubling the noodle length the most times, and in 2001, the Taiwanese cook Chang Hun-yu managed to double
his dough 14 times in two minutes. The noodle he ended up with was so thin that it could be passed through the eye of a needle and would have stretched from Mr. Chang’s restaurant in the center of Taipei to the outskirts of the city. When the noodle was cut, there were a total of 16,384 noodles.
This is the power of doubling, and it can very quickly lead to very big numbers. For example, if it were possible for Chang Hun-yu to have doubled his noodle 46 times, the noodle would have been the thickness of an atom and would have been long enough to reach from Taipei to the outer reaches of our solar system. Doubling the noodle 90 times would have gotten you from one side of the observable universe to the other. To get a sense of how big the current record prime number discovered in 2008 is, you would need to double the noodle 43,112,609 times and then take one noodle away.
WHAT ARE THE ODDS THAT YOUR TELEPHONE NUMBER IS PRIME?
One of the geeky things that mathematicians always do is check their telephone number to see whether it is prime. I moved homes recently and needed to change my telephone number. I hadn’t had a prime telephone number at my previous house (house number 53, a prime), so I was hoping that at my new house (number 1, an ex-prime), I might be luckier.
The first number the phone company gave me looked promising, but when I put it into my computer and tested it, I found that it was divisible by 7. “I’m not sure I’m going to remember that number . . . any chance of another number?” I asked. The next number was also not prime—it was divisible by 3. (An easy test to see whether your number is divisible by 3 is this: add up all the digits of your telephone number, and if the number you get is divisible by 3, then so is the original number.) After about three more attempts, the exasperated telephone company employee snapped: “Sir, I’m afraid I’m just going to give you the next number that comes up.” Alas, I now have an even telephone number, of all things!
So what were my chances of getting a prime telephone number? My number has eight digits. There is approximately a 1 in 17 chance that an eight-digit number is prime, but how does that probability change as the number of digits increases? For example, there are 25 primes under 100, which means that a number with two or fewer digits has a 1 in 4 chance of being prime. On average, as you count from 1 to 100, you get a prime every four numbers. But primes get rarer the higher you count.
This table shows the changes in probability:
Number of digits
Chance of getting a prime
1 or 2
1 in 4
3
1 in 6
4
1 in 8.1
5
1 in 10.4
6
1 in 12.7
7
1 in 15.0
8
1 in 17.4
9
1 in 19.7
10
1 in 22.0
Table 1.2
Though primes get rarer and rarer, they get rarer in a very regular way. Every time I add a digit, the probability decreases by about the same amount—2.3—each time. The first person to notice this was a 15-year-old boy. His name was Carl Friedrich Gauss (1777–1855), and he would go on to become one of the greatest names in mathematics.
Gauss made his discovery after being given a book of mathematical tables for his birthday. At the back of the book was a table of prime numbers. He was so obsessed with these numbers that he spent the rest of his life adding more and more figures to the tables in his spare time. Gauss was an experimental mathematician who liked to play around with data, and he believed that the way the primes thinned out would carry on in this uniform way however far you counted through the universe of numbers.
But how can you be sure that something strange won’t suddenly happen when you hit one hundred–digit numbers, or one million–digit numbers? Would the probability still be the same as adding on 2.3 for each new digit, or could the probabilities suddenly start behaving totally differently? Gauss believed that the pattern would always be there, but it took until 1896 for him to be vindicated. Two mathematicians, Jacques Hadamard and Charles de la Vallée Poussin, independently proved what is now called the prime-number theorem: that the primes will always thin out in this uniform way.
Gauss’s discovery has led to a very powerful model that helps to predict a lot about the behavior of prime numbers. It’s as if, to choose the primes, nature used a set of prime-number dice with all sides blank except for one with a big P on it for prime.
Figure 1.25 Nature’s prime-number dice.
To decide whether each number is going to be prime, roll one of the dice. If it lands prime side up, then mark that number as prime; if it’s blank side up, the number isn’t prime. Of course, this is just a heuristic model—you can’t make 100 indivisible just by the roll of dice. But it will give a set of numbers whose distribution is believed to be very like that of the primes. Gauss’s prime-number theorem tells us how many sides there are on each die. So for three-digit numbers, use a six-sided die or a cube with one side prime. For four-digit numbers, use an eight-sided die—an octahedron. For five digits, a die with 10.4 sides . . . of course, these are theoretical dice because there isn’t a polyhedron with 10.4 sides.
WHAT’S THE MILLION-DOLLAR PRIME PROBLEM?
The million-dollar question is about the nature of these dice: are the dice fair or not? Are the dice distributing the primes fairly through the universe of numbers, or are there regions where they are biased, sometimes giving too many primes, sometimes too few? The name of this problem is the Riemann hypothesis.
Bernhard Riemann was a student of Gauss’s in the German city of Göttingen. He developed some very sophisticated mathematics that allow us to understand how these prime-number dice are distributing the primes. Using something called a zeta function, special numbers called imaginary numbers, and a fearsome amount of analysis, Riemann worked out the math that controls the fall of these dice. He believed from his analysis that the dice would be fair, but he couldn’t prove it. To prove the Riemann hypothesis, that is what you have to do.
Another way to interpret the Riemann hypothesis is to compare the prime numbers to molecules of gas in a room. You may not know at any one instance where each molecule is, but the physics says that the molecules will be fairly evenly distributed around the room. There won’t be a concentration of molecules in one corner and a complete vacuum in another. The Riemann hypothesis would have the same implication for the primes. It doesn’t really help us to say where each particular prime can be found, but it does guarantee that they are distributed in a fair but random way through the universe of numbers. That kind of guarantee is often enough for mathematicians to be able to navigate the universe of numbers with a sufficient degree of confidence. However, until the million dollars is won, we’ll never be quite certain what the primes are doing as we count our way further into the never-ending reaches of the mathematical cosmos.
Two
THE STORY OF THE ELUSIVE SHAPE
The great seventeenth-century scientist Galileo Galilei once wrote, The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth.”
This chapter presents the A to Z of nature’s weird and wonderful shapes: from the six-pointed snowflake to the spiral of DNA, from the radial symmetry of a diamond to the complex shape of a leaf. Why are bubbles perfectly spherical? How does the body make such hugely complex shapes like the human lung? What shape is our universe? Math is at the heart of understanding how and why nature makes such a variety of shapes, and it also gives us the power to create new shapes, as well as the ability to say when there are no more shapes to be discovered.
It isn’t only mathematicians who are interested in shapes: architects, engineers, scientists, and artists all want to understand ho
w nature’s shapes work. They all rely on the mathematics of geometry. The ancient Greek philosopher Plato put above his door a sign declaring, “Let no one ignorant of geometry enter here.” In this chapter, I want to give you a passport to Plato’s home, to the mathematical world of shapes. And at the end, I’ll reveal another mathematical puzzle—one whose solution is worth another million dollars.
WHY ARE BUBBLES SPHERICAL?
Take a piece of wire and bend it into a square. Dip it in bubble mixture and blow. Why isn’t it a cube-shaped bubble that comes out the other side? Or if the wire is triangular, why can’t you blow a pyramid-shaped bubble? Why is it that, regardless of the shape of the frame, the bubble comes out as a perfect spherical ball? The answer is that nature is lazy, and the sphere is nature’s easiest shape. The bubble tries to find the shape that uses the least amount of energy, and that energy is proportional to the surface area. The bubble contains a fixed volume of air, and that volume does not change if the shape changes. The sphere is the shape that has the smallest surface area that can contain that fixed amount of air. That makes it the shape that uses the least amount of energy.
Manufacturers have long been keen to copy nature’s ability to make perfect spheres. If you’re making ball bearings or shot for guns, getting perfect spheres could be a matter of life and death, since a slight imperfection in the spherical shape could lead to a gun backfiring or a machine breaking down. The breakthrough came in 1783 when an English plumber, William Watts, realized that he could exploit nature’s predilection for spheres.