The Number Mysteries: A Mathematical Odyssey through Everyday Life. Marcus Sautoy du

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      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. Such is the power of doubling that the noodle would have stretched from Mr Chang’s restaurant in the centre of the Taipei to the outskirts of the city, and when it 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 Hunyu to have carried on and doubled his noodle 46 times, the noodle would be the thickness of an atom and would be long enough to reach from Taipei to the outer reaches of our solar system. Doubling the noodle 90 times would get you from one side of the observable universe to the other. To get a sense of how big the current record prime number is, you would need to double the noodle 43,112,609 times and then take one noodle away to get the record prime discovered in 2008.

      What are the odds that your telephone number is prime?

      One of the geeky things that mathematicians always do is to check their telephone number to see whether it is prime. I moved house 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?’ The next number was also not prime—it was divisible by 3. (An easy test to see whether your number is divisible by 3: 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 the chances of me 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.

      The table below shows the changes in probability:

      Primes get rarer and rarer, but 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 fifteen-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.

      TABLE 1.2

      Gauss made his discovery after being given a book of mathematical tables for his birthday which contained in the back 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 100-digit numbers, or 1,000,000-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 which helps to predict a lot about the behaviour 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 PRIME written on it:

      FIGURE 1.25 Nature’s prime number dice.

      To decide whether each number is going to be prime, roll 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 a roll of the 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 the dice. So for three-digit numbers, use a six-sided dice or a cube with one side prime. For four-digit numbers, an eight-sided dice—an octahedron. For five, digits a dice 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 which allows 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 mathematics 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 one corner with a concentration of molecules, and at another corner a complete vacuum. 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 certain quite what the primes are doing as we count our way further into the never-ending reaches of the mathematical cosmos.

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