The Amazing ABC Conjecture

In number theory, straightforward, reasonable questions are remarkably easy to ask, yet many of these questions are surprisingly difficult or even impossible to answer.

Fermat's last theorem, for instance, involves an equation of the form x^n + y^n = z^n. More than 300 years ago, Pierre de Fermat (1601-1665) conjectured that the equation has no solution if x, y, and z are all positive integers and n is a whole number greater than 2. Andrew J. Wiles of Princeton University finally proved Fermat's conjecture in 1994.

In order to prove the theorem, Wiles had to draw on and extend several ideas at the core of modern mathematics. In particular, he tackled the Shimura-Taniyama-Weil conjecture, which provides links between the branches of mathematics known as algebraic geometry and complex analysis.

That conjecture dates back to 1955, when it was published in Japanese as a research problem by the late Yutaka Taniyama. Goro Shimura of Princeton and Andre Weil of the Institute for Advanced Study provided key insights in formulating the conjecture, which proposes a special kind of equivalence between the mathematics of objects called elliptic curves and the mathematics of certain motions in space.

The equation of Fermat's last theorem is one example of a type known as a Diophantine equation -- an algebraic expression of several variables whose solutions are required to be rational numbers (either whole numbers or fractions, which are ratios of whole numbers). These equations are named for the mathematician Diophantus of Alexandria, who discussed such problems in his book Arithmetica.

In fact, it was in the margin of a page of a Latin translation of Arithmetica that Fermat first set down the proposition that came to be known as Fermat's last theorem. He had studied the book closely, making marginal notes in his copy. After Fermat's death, his son published a new edition of Arithmetica that included the notes in an appendix.

Interestingly, the Wiles proof of Fermat's last theorem was a by-product of his deep inroads into proving the Shimura-Taniyama-Weil conjecture. Now, the Wiles effort could help point the way to a general theory of three-variable Diophantine equations. Historically, mathematicians have always had to state and solve such problems on a case-by-case basis. An overarching theory would represent a tremendous advance.

The key element appears to be a problem termed the ABC conjecture, which was formulated in the mid-1980s by Joseph Oesterle of the University of Paris VI and David W. Masser of the Mathematics Institute of the University of Basel in Switzerland. That conjecture offers a new way of expressing Diophantine problems, in effect translating an infinite number of Diophantine equations (including the equation of Fermat's last theorem) into a single mathematical statement.

Like many problems in number theory, the ABC conjecture can be stated in relatively simple, understandable terms. It incorporates the concept of a square-free number: an integer that is not divisible by the square of any number. For instance, 15 and 17 are square-free, but 16 and 18 are not.

The square-free part of an integer n is defined to be the largest square-free number that can be formed by multiplying the prime factors of n. That quantity is denoted sqp(n). Thus, for n = 15, the prime factors are 5 and 3, and 3 x 5 = 15, a square-free number. So sqp(15) = 15. On the other hand, for n = 16, the prime factors are all 2, which means that sqp(16) = 2. Similarly, sqp(17) = 17 and sqp(18) = 6.

In general, if n is square-free, the square-free part of n is just n. Otherwise, sqp(n) represents what's left over after all the factors that create a square have been eliminated. In other words, sqp(n) is the product of the distinct prime numbers that divide n. So sqp(9) = sqp(3 x 3) = 3; sqp(1400) = sqp(2 x 2 x 2 x 5 x 5 x 7) = 2 x 5 x 7 = 70.

With these preliminaries out of the way, mathematician Dorian Goldfeld of Columbia University describes the ABC conjecture in the following terms: The problem deals with pairs of numbers that have no factors in common. Suppose A and B are two such numbers and that C is their sum. For example, if A = 3 and B = 7, then C = 3 + 7 = 10. Now, consider the square-free part of the product A x B x C: sqp(ABC) = sqp(3 x 7 x 10) = 210.

For most choices of A and B, sqp(ABC) is greater than C, as in the example above. In other words, sqp(ABC)/C is larger than 1. Once in a while, however, that isn't true. For instance, if A is 1 and B is 8, then C = 1 + 8 = 9, sqp(ABC) = sqp(1 x 8 x 9) = sqp(1 x 2 x 2 x 2 x 3 x 3) = 1 x 2 x 3 = 6, and sqp(ABC)/C = 6/9 = 2/3. Similarly, if A is 3 and B is 125, the ratio is 15/64, and if A is 1 and B is 512, the ratio is 2/9.

Masser proved that the ratio sqp(ABC)/C can get arbitrarily small. In other words, if you name any number greater than zero, no matter how small, you can find integers A and B for which sqp(ABC)/C is smaller than that number.

In contrast, the ABC conjecture states that [sqp(ABC)]^n/C does reach a minimum value if n is any number greater than 1 -- even a number such as 1.0000000000001, which is just barely larger than 1. The tiny change in the expression makes a vast difference in its mathematical behavior.

Astonishingly, a proof of the ABC conjecture would provide a way of establishing Fermat's last theorem in less than a page of mathematical reasoning. Indeed, many famous conjectures and theorems in number theory would follow immediately from the ABC conjecture, sometimes in just a few lines.

"The ABC conjecture is amazingly simple compared to the deep questions in number theory," says Andrew J. Granville of the University of Georgia in Athens. "This strange conjecture turns out to be equivalent to all the main problems. It's at the center of everything that's been going on."

"Nowadays, if you're working on a problem in number theory, you often think about whether the problem follows from the ABC conjecture," he adds.

"The ABC conjecture is the most important unsolved problem in Diophantine analysis," Goldfeld writes in Math Horizons. "It is more than utilitarian; to mathematicians it is also a thing of beauty. Seeing so many Diophantine problems unexpectedly encapsulated into a single equation drives home the feeling that all the subdisciplines of mathematics are aspects of a single underlying unity, and that at its heart lie pure language and simple expressibility."

It comes as no surprise that mathematicians are hard at work searching for a route to conquer the remarkable ABC conjecture.

References:

Goldfeld, Dorian. 1996. Beyond the last theorem. Math Horizons (September):26-34.

______. 1996. Beyond the last theorem. The Sciences (March/April):34-40.

Guy, Richard K. 1994. Unsolved Problems in Number Theory. New York: Springer-Verlag.

Peterson, Ivars, 1997. Prize offered for solving number conundrum. Science News 152(Nov. 15):310. Article available at http://www.sciencenews.org/sn_arc97/11_15_97/.fob2.htm.

______. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.

______. 1988. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W.H. Freeman.

Singh, Simon, and Kenneth A. Ribet. 1997. Fermat's last stand. Scientific American (November):68-73.

Wiles, Andrew. 1995. Modular elliptic curves and Fermat's last theorem. Annals of Mathematics 141(May):443-551.

Additional information about the ABC conjecture is available at http://www.astro.virginia.edu/~eww6n/math/abcConjecture.html.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.