Most children are able to deal with integer numbers in kindergarten. Manipulating fractions is a bit more difficult. The dear little ones will have had to attend primary school for a couple of years in order to handle them. But irrational numbers are a different matter altogether. Dealing with numbers that cannot be expressed as a fraction of two whole numbers is where the real difficulties start.

The exact opposite holds true for equations. It is fairly easy to find irrational solutions to problems. The real rub starts when a problem requires that the solutions be only integer numbers. The section of mathematics that deals with such problems is called number theory. An annoying characteristic of the discipline is its apparent simplicity. At first glance, statement of the problems seems easy enough. It is only when one delves into it more deeply that the horrendous difficulties become apparent.

The Greek mathematician Diophantus, who lived some 1,800 years ago in Alexandria and is often known as the father of algebra, is said to have founded number theory. In his honor, equations with unknowns that must be integer numbers are called Diophantine equations.

Diophantus’ main work, the Arithmetika, consisted of some 130 problems and their solutions. Unfortunately, the books were destroyed during a fire in the small library of Alexandria in the year 391. Many years later, in the 15th century, six of the original 13 books were discovered. (Then, in 1968, another four volumes surfaced, albeit in an incomplete Arabic translation.) For years people puzzled over the manuscripts of the ancient Greek mathematician and only in the 17th century was someone finally able to get a handle on the material. This man was Pierre de Fermat, a French magistrate who enjoyed spending his spare time playing around with mathematics. To-

day Fermat is known above all for his notorious Last Theorem. (See also Chapter 28.)

One problem that originated with Diophantus remains unsolved to this very day: Which numbers can be expressed as the sum of two integer numbers or fractions, each of them raised to the third power? The question can be answered in the affirmative with the numbers 7 and 13, for example, since 7 = 2³ + (−1)³ and 13 = (7/3)³ + (2/3)³. But what about numbers like 5 or 35? To answer the question, one needs to be familiar with the most complicated methods in modern mathematics.

All that mathematicians have found so far is a method to determine whether a decomposition of a specific number exists. But they are unable to provide the decompositions themselves. To determine whether a number can be decomposed into cubes, the graph of a so-called L function must be calculated for this number. If the graph intersects or touches the x axis of the coordinate system at precisely the point x = 1, the number in question can be decomposed into cubes. If the value of the function at x = 1 is not 0, no decomposition is possible. The condition is met for the number 35: The L function associated with it becomes 0 at exactly x =1. Indeed, 35 can be decomposed into 3³ + 2³. On the other hand, for the number 5, the graph of the L function neither touches nor intersects the x axis. This proves that 5 cannot be decomposed into cubes.

Don Zagier, the director of the Max Planck Institute of Mathematics in Bonn, Germany, gave two lectures to the general public in 2003 in Vienna about Diophantine cubic decompositions. Zagier is one of the world’s leading mathematicians, and his main area of work is number theory. As a child he was already known to be a wunderkind. Born in the German city of Heidelberg in 1951, he grew up in the United States, finished high school at age 13, completed undergraduate studies in mathematics and physics at the Massachusetts Institute of Technology at 16, and gained a Ph.D. from Oxford at 19. By the age of 23 he obtained the habilitation—the German qualification to teach as a professor—at the Max Planck

Institute of Mathematics. At the age of 24 he was the youngest professor in all of Germany. His talents are not limited to mathematics, by the way: He speaks nine languages.

One of Zagier’s talks, part of the Gödel lecture series in Vienna, was entitled “Pearls of Number Theory.” The other lecture was held at the opening of “math.space,” a unique hall in Vienna’s museum quarter whose purpose it is to host popular presentations on mathematics. The hope is that this allegedly esoteric topic could be made accessible to the city’s wider public who usually pass their time at operas and in coffeehouses.

Zagier is a quirky little man. But when he starts explaining his pet theory to the audience his performance would make a rock star pale in envy. Constantly jumping back and forth between two overhead projectors, he enthralls his audience with mathematical explanations, delivered in perfect German albeit tinged with an American accent. Even a fierce mathematophobe would forget that he is listening to a math lecture. The joy that Zagier—known by some as Bonn’s superbrain—feels for his vocation is obvious to all. Watching him is like watching a concert virtuoso. It is hard to believe that mathematicians, such as Zagier, are sometimes accused of being involved in a dry sort of science.