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In 1859 Bernhard Riemann, a shy German mathematician, wrote an eight-page article giving an answer to a problem that had long puzzled mathematicians. But he didn’t provide a proof. In fact, he said he couldn’t prove it but he thought that his answer

- ISBN-13:
- 9780374250072
- Publisher:
- Farrar, Straus and Giroux
- Publication date:
- 03/19/2003
- Edition description:
- 1 AMER ED
- Pages:
- 304
- Product dimensions:
- 5.82(w) x 8.48(h) x 1.15(d)

All right reserved.

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. -Leonhard Euler

The Riemann Hypothesis emerged from the attempts of mathematicians to understand the subtleties of prime numbers. These are whole numbers, also called integers, that cannot be divided by smaller integers without leaving a remainder. They are fundamental to our number system, as Jon Keating, a mathematician working at the University of Bristol, explains:

"Primes are like pieces of Lego. You have individual blocks of Lego which you can't break down any further. The smallest blocks of Lego come in different sizes but you can't break them in half. They're the primes. Out of those blocks you can build buildings, you can build Lego objects. Those are like all the other numbers, the nonprimes. One question is, is a brick itself a Lego object? I would say yes, and therefore in that sense the primes are different from the rest of the numbers, but they are part of the numbers-they are the numbers you can build everything out of and they're the numbers you can't break down any further."

Prime numbers are the ones printed in bold in the following list. It's a list with which everyone is familiar-the counting numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, ...

The significance of the list is less in the primes than in the other numbers. All the others can be obtained by multiplying together some combination of prime numbers from earlier in the list. For example, 6 is 2 x 3. But you can't get any of the prime numbers by multiplying other numbers together. You can't "split" 19 into the product of smaller numbers ("product" in mathematics meaning the result of a particular multiplication).

There is a chemical analogy that might help to underline the importance of prime numbers. It involves the difference between atoms and molecules. Atoms are the basic building blocks of all matter in the universe. There are 92 naturally occurring elements, from hydrogen to plutonium, each with a characteristic type of atom. These atoms have the ability to stick together, either, in collections of atoms of the same element-say lots of carbon atoms joined together-or of different elements-hydrogen and oxygen, say. A collection of atoms-whether the same or different-linked firmly together is called a molecule. There are many different types of molecules, far more than 92. Although not infinite in number, molecules come in so many different shapes and sizes-because 92 types of atoms can combine in many different ways-that for practical purposes they are uncountable.

Any substance, from wood to water, cheese to chalk, can be described by a formula which indicates the exact atomic makeup of each molecule of the specific substance. To get a molecule of water, you take two atoms of hydrogen and attach them to one atom of oxygen, and so the molecule is described as [H.sub.2]O. Larger molecules are more complex. The mineral called marialite has the formula N[a.sub.4]A[l.sub.3]S[i.sub.9][O.sub.24]Cl, which means that a molecule of marialite is made up of four atoms of sodium (Na) joined to three of aluminum, nine of silicon, twenty-four of oxygen, and one of chlorine.

Now, the whole numbers, sometimes called the counting numbers, that we all use every day also fall into two types, a little like atoms and molecules. The trouble is, we don't usually realize this because the two types are mixed up. It's as if water, salt, and hemoglobin (molecules) were included in the same list as hydrogen, carbon, and iron (atoms). The two types of whole number are the prime numbers and the nonprimes, also called composite numbers, just as hydrogen could be seen as "prime" and water as "composite."

In both cases, each composite member is a unique arrangement of the units that make it up. Just as a specific molecule, say of sulfuric acid, is made up of a unique set of atoms-[H.sub.2]S[O.sub.4], or two hydrogens, one sulfur, and four oxygens-so a number that is not prime, such as 108,045, is made up of a unique grouping of prime numbers-in this case two 3's, one 5, and four 7's multiplied together, which you could call [3.sup.2] x 5 x [7.sup.4], almost like a formula for this particular composite number. There is no other combination of atoms that will make sulfuric acid, and there is no other combination of primes that will make 108,045. This fact about the whole numbers, which can be proved to be true, is known as the Fundamental Theorem of Arithmetic. But there is one important difference: the atoms run out before you get to a hundred, whereas the primes go on forever. However large a prime number you discover, I can always show you a larger one.

The infinity of the primes was one of the earliest facts to be established about these special numbers. Since the first known proof, by Euclid in the third century BC, mathematicians have come up with hundreds of important statements about subtler aspects of these numbers, based on analyzing how they are actually spaced out along what's called the number line, which stretches from its starting point at zero toward infinity.

Imagine the numbers that we are all familiar with, laid out in a straight line from zero to infinity. You could visualize it as a long straight road with a neat line of houses continuing into the distance, each house numbered in order from 1 to infinity. Now, some houses have one person living in them, and others more than one. The one-person houses are "primes," and their addresses are prime numbers. The multi-occupant houses have people living in them who are all related to someone living in a prime house. There's a simple rule that determines which family members are living where. If the address is divisible by the address of a "prime" house, then there's someone from that family living there. For example, house number 6 has two people in it: one from the same family as the resident of number 2 and one related to the inhabitant of number 3.

Number 3 has an address that is not divisible by any other address, so it just has one person, a member of the 3 family living there. And number 4 has two members of 2 living in it, because 4 is divisible by 2 twice, since 4 is 2. But number 12 has people from the 2 family and the 3 family living there-two members of 2 and one of 3.

It may seem a cumbersome analogy, but it does have its uses. Along the number street every house has at least one inhabitant, and as you go along the street the houses have to accommodate more and more people because their addresses are divisible by more and more primes. But it turns out that as you walk along the road, passing houses with more and more people living in them, there are still occasional one-person houses scattered here and there. However far you travel, there seem to be members of new families ahead, new primes living in solitary comfort.

As mathematicians have explored this strange road they've found a lot of surprising things. For example, there doesn't seem to be a regular pattern to the distribution of the primes, though there are certain regularities. There is never a situation-apart from houses 2 and 3-where two primes live next door to each other. But there are many instances-in fact there are believed to be an infinite number-of two primes separated by one multi-occupant house. This, known as the Twin Primes conjecture, is just one of many hypotheses in number theory which are simple to state but have resisted all attempts at a proof.

There are some long stretches of multi-occupancy houses with no primes at all. In fact, you would find that however long a stretch you care to suggest (a hundred houses, a thousand houses, ...), somewhere along this road will be a stretch of multi-occupancy houses as long as that, uninterrupted by primes. There doesn't seem to be a rule about how many people live in adjacent houses. The sole resident of number 10,007 (a prime) lives next door to number 10,006, containing one person from family 2 and one from family 5,003; and next to them is a house, number 10,005, that is home to four residents, one each from families 3, 5, 23, and 29.

Looking at the straight line of whole numbers, the primes get farther apart as you move along the line toward infinity. This suggests that you might eventually reach a number that is the last prime number-from then on, perhaps, every number can be obtained by multiplying together some combination of lower numbers. But in fact, that doesn't happen. However far you go along the road there will always be more primes ahead, even if they are separated more and more by multi-occupancy houses. Since Euclid's proof of this fact, there have been many different proofs of the infinity of the primes. Let's look at one of them.

Suppose we assume that there *is* a largest prime number-call it *L*. If we use that assumption as a starting point, and then, by a series of logically impeccable steps, find a way to create a prime number larger than *L*, this would show that our initial assumption must have been wrong. If I say "all roads lead to Rome," set off down any old road, and arrive in Florence, I have *proved* that "all roads lead to Rome" is a false statement. So if we can show a way of making a prime number larger than *L*, this will show that there is no number *L* that is the largest prime number. In doing this we will have experienced a particular type of proof called a "reductio ad absurdum." Here's one way it works:

Starting with our assumed largest prime, *L*, we multiply it by all smaller prime numbers to get a number-call it ITLITL-that is clearly not prime but composite, since it can be divided by any of the smaller numbers. But let us now add 1 to this composite number. Now, is this new number, ITLITL + 1, prime or composite? If it's prime, there must be no numbers that divide it exactly. The way to discover if this is the case is to divide it one by one by all smaller prime numbers. Now, every time we divide ITLITL + 1 by a smaller prime we get an answer that is not a whole number. To make this clear, suppose someone said that 17 is the largest prime. If we multiply 17 and all smaller primes together we get 2 x 3 x 5 x 7 x 11 x 13 x 17 = 510510. Add 1 and we get 510511. So no primes equal to or less than 17 will divide into it exactly. Try dividing by 7 for example:

2x3x5x7x11x13x17+1/7 = 2x3x5x7x11x13x17+1/7

So ITLITL + 1 must either be a prime, and one that is larger than *L*-thus contradicting our initial assumption that L is the largest prime-or it must be divisible by a prime larger than *L*, in which case again *L* is not the largest prime. To recap, having assumed there was a largest prime number, *L*, we produced a contradiction-a prime number that was larger still, either ITLITL + 1 or a divisor of it that is larger than *L*. And the method we've used would apply to any suggested largest prime number. So this means that our original assumption was wrong-there is no largest prime number; they just go on and on.

Of course, this doesn't mean there isn't a largest *known* prime number. There will always be one of those, for as long as people with time on their hands choose to work out a larger one or, these days, choose to program ever more powerful computers to do the search. In 1876, the new record-holder became

170, 141, 183, 460, 469, 231, 731, 687, 303, 715, 884, 105, 727

It has 39 digits and remained the largest prime until, in 1951, thanks to a roomful of valves and circuits that comprised one of the earliest digital computers, a larger prime number was discovered. This one had 44 digits, but it didn't hold the record for long, as prime numbers with 79, 157, and then several hundred digits were discovered.

According to a news clipping (shown in Figure 1) which turned up among the papers of the American mathematician Julia Robinson, a larger prime was discovered by a Chicago mathematician. There's something odd about this clipping. The number Dr. Krieger found is larger than the 44-digit discovery made using the primitive computer, and yet it appears he did this merely with pencils and paper. It turns out, as Andrew Odlyzko has pointed out to me, that this number is not prime-it is divisible by 7. The only other fact I can find out about Dr. Krieger is that in 1938 he reported a counterexample to Fermat's Last Theorem, an example that proved the theorem wrong. More than fifty years later, the English mathematician Andrew Wiles proved the theorem correct, so Dr. Krieger's record as a mathematician is not very impressive.

Since the early days of computers, finding large primes has become little more than a harmless pastime for anyone with a PC. The search has lost its glamour, and larger ones are emerging all the time. In 1998, Roland Clarkson, a nineteen-year-old California student, discovered a new prime, [2.sup.3,021,377] - 1, which has 909,526 digits. A mass computer project called GIMPS (Great Internet Mersenne Prime Search) makes use of the downtime on thousands and thousands of PCs around the world, and the largest known prime number, as I write, is [2.sup.13,466,917] - 1. It has more than 4 million digits, and if you printed it out in type this size, it would be three miles long.

For some mathematicians, even those passionately interested in primes, the very idea of devoting time to finding the largest prime at any one time is not worth the effort.

"Mathematically this has almost no interest," said Andrew Granville of the University of Georgia. "I would be recognized as one of the world experts on such techniques and I'm very interested in how you would go about finding a very large prime number, but it's almost irrelevant when you're doing it except that it can be done. But what happens is every time somebody finds a new largest prime number, it's a cheap market for them. They figure people like to hear this stuff. For people who like a little math in their lives it really cheapens the subject because it's not very relevant, the actual calculations. There are some calculations that are fantastic and well worth knowing about, but finding the largest prime number ...?"

There's a retired engineer called Harvey Dubner who would disagree profoundly with Granville's view. He spends his days finding ever more curious and unusual primes just for the sake of it, like a lepidopterist looking for more and more exotic butterflies. To write these primes, a new notation is needed to save ink and paper. *Continues...*

Excerpted fromThe Riemann HypothesisbyKarl SabbaghCopyright © 2002 by Karl Sabbagh

Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

**Karl Sabbagh** is the author of six books, most recently* A Rum Affair *(FSG, 2000). He lives near Stratford-upon-Avon in England.

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