Sometimes work in one discipline of pure mathematics has a completely unexpected payoff in another. Some of the famous mathematician Pierre de Fermat’s (1601–1665) work in number theory bears this out. It took 150 years, however, until the mathematician Carl Friedrich Gauss (1777–1855) found a geometric application to one of Fermat’s statements in number theory: the construction of regular polygons by ruler and compass.

Fermat is famous not only for his notorious “Last Theorem,” which had actually been no more than a conjecture until it was finally proven by Andrew Wiles in 1994. As a magistrate in the French city of Toulouse, a position he held throughout his adult life, Fermat was apparently not being kept too busy. His post left him sufficient time to pursue his mathematical passions. In correspondence with Marin Mersenne, a monk who shared Fermat’s love for mathematics, he discussed problems in number theory. Mersenne was preoccupied at the time with numbers of the form 2ⁿ + 1, and Fermat conjectured that, if n is a power of 2, the numbers are always prime. Ever since, numbers of the form 2²ⁿ + 1 are called Fermat numbers.

Fermat himself did not provide any proof for this conjecture. (In fact, many of his proofs are lost, and it is possible that some of them were not rigorous. Deductions by analogy and the intuition of genius sufficed to lead him to correct results.) Of the Fermat number he knew only the zeroth and the four subsequent ones (3, 5, 17, 257, 65537). The next Fermat number, 2³² + 1, was too large to be computed in his time, let alone be checked for primality. But the first five Fermat numbers are, indeed, divisible only by 1 and by themselves. To conclude from this that all such numbers are prime would have

been a rather daring leap, however. And indeed, this was one case where one of Fermat’s conjectures was wrong, which may have a rather sobering effect on us mortals: Not all conjectures made by famous mathematicians are necessarily correct.

It was nearly a century, however, before Leonhard Euler of Basle, Switzerland, provided a counterexample. In 1732 he showed that the Fermat number corresponding to n = 5 (which equals 4,294,967,297) is the product of the numbers 641 and 6,700,417. Hence not all Fermat numbers are prime. But which are and which are not?

The search for the answer is still going strong. In 1970 it was shown that the Fermat number corresponding to n = 6 is also a composite number. Today, volunteers all over the world offer their PCs’ idle time to test for the primality of Fermat numbers. In October 2003 it was announced that the Fermat number —a number so unimaginably huge that writing it down would require a blackboard thousands of light years long—is composite.

Unfortunately, there are large gaps in the list of tested numbers. In fact, a mere 217 of the first 2.5 million Fermat numbers have been tested so far. Contrary to Fermat’s prediction, not one of them—with the exception of the first five—has turned out to be prime. The failure to come up with any more Fermat numbers that are prime led to a new conjecture—the direct opposite of Fermat’s original one: All Fermat numbers, excepting the first five, are composite. This new conjecture remains, like the old one, without proof. Nobody really knows whether there are more than five prime Fermat numbers, if there exist an infinite number of composite Fermat numbers, or if all Fermat numbers, excepting the first five, are prime.

And now to the geometric application.

In 1796 Carl Friedrich Gauss, a 19-year-old student at the University of Göttingen, was thinking about which regular polygons could be constructed using only ruler and compass. Euclid had, of course, been able to construct regular triangles, squares, and pentagons already. But 2,000 years later mankind had not gotten much further than that. What about the regular 17-cornered poly-

gon? To his immense satisfaction, the young Gauss was able to prove that the so-called heptadecagon can actually be constructed. But he did still more. Gauss proved that every polygon whose number of corners is equal either to a prime Fermat number or to the product of prime Fermat numbers is constructible by ruler and compass. (To be precise, this also holds when the number of corners is doubled and redoubled, since angles can always be halved with the help of ruler and compass.)

It follows that it is also possible to construct a 257-cornered polygon, corresponding to the next Fermat number, and instructions as to how to go about constructing a regular 65,537-cornered polygon, on which a certain Johann Gustav Hermes spent 10 years of his life, are tucked away in a box that today is safeguarded in the library of the University of Göttingen.

Gauss suspected that this statement also holds in the opposite direction: The number of corners of any polygon that is constructible by ruler and compass must be a product of Fermat numbers. Indeed, this time the conjecture was correct. It was not Gauss, however, who proved this. The honor fell to Pierre Laurent Wantzel, a French mathematician, who presented a proof in 1837.

During his lifetime, Gauss made innumerable mathematical discoveries. Yet he considered the one involving the construction of the 17-cornered polygon one of the most significant. So highly did he value the discovery of his youth that he expressed the wish to have an image of this figure carved onto his tombstone. The stonemason, thus goes the story, refused, claiming that a 17-cornered polygon would be much too close to a circle. Eventually, a monument was commissioned by the city of Brunswick, Gauss’s birthplace, whose pillar was decorated by a 17-pointed star.