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A Pure Love of Math - The Riemann Hypothesis

Enrico Bombieri is one of a small but dedicated band of mathematicians who over the last 150 years have labored over the proof of the Riemann Hypothesis, an extraordinarily difficult problem that interweaves the arithmetic properties of the counting numbers with advanced analysis. While hard to understand in detail, it is easy to see why this problem fascinates mathematicians as a candle flame does a moth.

Problem solving is the stuff of mathematics. And its most famous to-do list was that of the great German mathematician, David Hilbert, who believed passionately that every problem could be solved. In a Paris lecture in August, 1900 he named the ten most important mathematical problems to be solved in the 20th century. The Riemann Hypothesis was problem four. (His list grew to 23 problems in later published papers).

The Riemann Hypothesis describes the behavior of particular values of a specific mathematical recipe or function, called the zeta function. These can, in turn, be linked with the properties of the prime numbers -- those whole numbers that have no divisors other than themselves and 1. Every calculated example supports the hypothesis and it is widely believed to be true. And yet, after one hundred years of effort, no one has yet fulfilled Hilbert's demand for a proof. It remains a holy grail of pure mathematics.


Why would anyone dedicate their life to the study of a problem so difficult that generations of mathematicians have failed to solve it? The answer may lie in the challenge itself. Before 1953, Mt. Everest represented an essential challenge. Although the tallest mountain on Earth, no climber had stood on its summit. Many aspired to be first, to do what none had done. Nearly 50 years after Edmund Hillary and Tenzing Norgay first reached the top, Everest remains a test and a goal for the best of mountaineers. For mathematicians, an especially difficult unsolved problem is an unreached peak, an intellectual summit as yet unvisited.

The analogy is, of course, imperfect. Standing at its base, you can see a mountain's summit far above you. You can plan in detail the route you will attempt to reach it. You know in advance the difficulties of terrain, of objective danger, and of technique you will face. What you do not know is whether you will overcome them and reach the summit.

The quest for an unproven mathematical result differs. Though you can glimpse the result you seek, the terrain below is shrouded in fog and mist, the best mathematical direction to approach the summit unknown. You may not even know the techniques you will need to assist your climb. If your hoped-for result proves false, there may not even be a summit at all. Instead of falling from a high perch, you risk seeing the mathematical mountain itself collapse in a rubble of dashed hopes and expectations.

Of Hilbert's summits still unclimbed, the Riemann Hypothesis is one of the most tantalizing, in part because its association with number theory is simple and pure. And because number theory starts from our common understanding of numbers and arithmetic, we can appreciate the excitement of the ascent.

Our route starts with a bag of six ordinary marbles and a simple property of numbers -- their divisibility. You can split the marbles evenly into two equal groups of three each: "Six divided by two equals three." Add more marbles in the bag. Which numbers of marbles can be evenly divided, and which not?

Early in school, we discover that the numbers 2, 4, 6, 8 ... divide in half, while 1, 3, 5, 7 ... do not. "Even" and "odd" are mathematical descriptions. And if you can divide a particular bag of marbles into two equal bunches, then when you add one more marble to the bag, you can no longer do so. Even and odd numbers alternate!

What of other divisions? A bag with 3, 9, or 12 marbles can be split into three equal bunches: 5 and 7 cannot. So are there bags of marbles that cannot be divided in any way into equal multiple bunches? Bags with 3, 5, 7, 11, 13, 17 ... marbles prove to be indivisible. These numbers (including 1 and 2 as well) that are evenly divided only by themselves and one we call prime numbers. How does one tell if a given number is prime? Try every possible divisor, 2, 3, 5, 7 ... in turn. If you reach the number itself, with no joy in division, you have proven your number is prime.

Doing nothing fancier than splitting a bag of marbles, we have discovered questions worth asking. Are there any even prime numbers greater than 2? No, since every even number is divisible into two equal parts! Do primes grow larger without limit or is there a largest prime number, such that every larger number is divisible? This conjecture, or good mathematical guess, was a significant math summit reached by Euclid. Starting by assuming such a number did exist, he was able by contradiction to prove that prime numbers grow larger than any number we can name.



What if we could pour numbers through a kind of sieve that caught each prime number while letting all the divisible numbers fall through? The method, ascribed to the Greek scholar Eratosthenes, is simplicity itself: form a table of numbers and strike out all multiples. Write the whole numbers from 1 to 50 in a table. Then strike out every multiple of 2 in the table, since none can be prime. And then strike out every multiple of 3, for the same reason. Then five, then seven, and so on.

All that can be left in the sieve are numbers less than 50 with no divisors. The sieve contains every prime less than 50! While the method may seem tedious, this 2,200-year-old method is simple and powerful enough to be used by many computer programs to find primes!

What patterns can we find among the prime numbers in our sieve?


The primes appear helter-skelter among the other numbers with no apparent pattern to their location. This random position means that we cannot calculate every prime by a fixed formula, or predict the exact location of a prime number. No way exists to find them all but arithmetic combined with educated guesswork.

The mathematician asks, "How can this be? Primes must follow set rules. There must be patterns in their placement." And so they seek to discover such patterns, roaming distant mathematical mountain ranges in search of unclimbed prime summits.

Not all these summits are distant, or remote. Look at number coincidences in the list of primes. They occasionally pair up, like 3 and 5; 5 and 7; or 11 and 13. These primes are twins, nearest odd number neighbors. They seem harder to find as we examine ever larger prime numbers in our sieve. And so a conjecture: do twin primes also continue without limit or do they eventually run out so that very large prime numbers are always found alone? A simple question, as yet unanswered.

One good pattern leads to another. What about prime triplets, numbers like 3, 5, 7? How many of these are there? Not a mountain, but a mathematical molehill: there are no other triplets. Multiples of three in the sieve form a picket fence that prevents such triplets. A picture shows why:


A prime triplet consists of three successive odd numbers. But only one of the pairs can straddle the picket of multiples; the other matches a multiple of three and so can't be a prime. A modest result, perhaps, but a summit nevertheless. No prime triplets! And the proof evokes a bit of the excitement of venturing into prime territory, a bit of the thrill of reaching unknown summits.

Our final arithmetic summit is a pattern first recognized by the mathematicians Goldbach and Euler in 1742. Take a bag of 8 marbles and split them into two unequal piles, 5 in one and 3 in the other. 8 = 5 + 3. Both prime numbers. Can we write every number as a sum of two primes? Certainly not, since primes larger than 2 are odd and "two odds make an even." But perhaps you can form every even number greater than 2 by adding together exactly two primes. Prove this Goldbach conjecture and many mathematicians will want to hear the story of your climb!

So, mathematicians who dare to work at a high mathematical altitude may share with high climbers an emotional and adrenal "edge" that fuels their efforts. And strive for that unique moment of proof when they are first to know a new truth to a certainty. What honor, what prize, what earthly reward could possibly compete with the satisfaction of having solved one of mathematics' great problems?

-- Michael Templeton


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