Not Proved

There was a flurry of excitement among mathematicians recently when it seemed for a few weeks that Fermat’s Last Theorem, an old and venerable chestnut among number theorists, had finally been proved. Alas, the proof proposed by Yoichi Miyaoka by way of algebraic geometry has been found lacking by the experts. Fermat’s Last Theorem, which has tantalized mathematicians for 350 years, will elude them at least a little while longer. How much longer is anyone’s guess.

Fermat’s Last Theorem, proposed by Pierre de Fermat around 1637, asserts that there is no perfect cube that is the sum of two perfect cubes, no fourth power that is the sum of two fourth powers, no fifth power that is the sum of two fifth powers, and so on forever. (There are, however, infinitely many perfect squares that are the sum of two perfect squares. Five squared (25) is the sum of three squared (9) plus four squared (16), for example.)

Fermat asserted his theorem in the margin of a book he was reading, adding, “I have found a truly marvelous proof of this theorem but this margin is too narrow to contain it.” This statement “is as famous in the history of mathematics as, say, the French Revolution is in the history of modern man,” Edna E. Kramer wrote in “The Nature and Growth of Modern Mathematics.”

To date, no one has been able to prove Fermat’s Last Theorem (it is the last because all of his others have been proved) or to disprove it by offering a counter-example. So when an established and respected number theorist offered what seemed to be a plausible proof, people got very excited. But after examining his proof, the experts concluded that while Miyaoka’s attack might yet work, it hasn’t worked yet.

A mathematician we know was wondering whether a proof of Fermat’s Last Theorem would mean “the end of mathematics,” by which he meant that only expert mathematicians could understand all the remaining problems.

Fortunately, that is clearly not the case. While there are many questions that require advanced training in mathematics, there are many simple-sounding problems in number theory still to be answered. Many of them were known to the Greeks.

For example, twin primes are prime numbers (which are divisible only by themselves and 1) that differ by 2, such as 3 and 5, 5 and 7, 11 and 13, and 17 and 19. Are there infinitely many such pairs? No one knows.

Another unsolved problem is the Goldbach Conjecture, which asserts that every even number greater than 2 can be written as the sum of two primes: 4 equals 2 + 2; 6 equals 3 + 3; 8 equals 3 + 5; 10 equals 3 + 7. This statement has never been proved; nor is there a counter-example.

A perfect number is a number that is the sum of its divisors other than itself (6 equals 1 + 2 + 3; 28 equals 1 + 2 + 4 + 7 + 14). There are 31 such perfect numbers known. Are there infinitely many of them? No one knows.

So we trust that there is plenty for mathematicians to do. Besides, Fermat’s Last Theorem is out there to be proved. Miyaoka’s valiant effort has given mathematics the Miyaoka Conjecture, which, if true, would imply Fermat’s Last Theorem. A solution to the problem may be near, or Miyaoka’s approach may prove a dead end. It wouldn’t be the first time.

At the start of this century, the great mathematician David Hilbert declared, “We must know. We will know.” A few decades later, the logician Kurt Godel amended it to say, “Well, maybe, maybe not.”

Perfect knowledge, even in mathematics, can never be achieved. Fermat’s Last Theorem may be a true statement that cannot be proved.