The curious power of large numbers.

The curious power of large numbers

Sometimes it takes more than 200 years and the help of a computer to solve a mathematics problem. In 1769, Leonhard Euler, while thinking about a problem now known as Fermat's last theorem (SN: 6/20/87, p.397), proposed that no set of positive integers, a, b, c and d, satisfies the equation a.sup.4 + b.sup.4 + c.sup.4 = d.sup.4 in the same way that numbers such as 3, 4 and 5 satisfy the more familiar equation x.sup.2 + y.sup.2 = z.sup.2.. Euler's guess seemed reasonable because Pierre de Fermat, a century earlier, had proved that the simpler equation a.sup.4 + b.sup.4 = c.sup.4 had no positive integer solutions.

Last summer, mathematician Noam D. Elkies of Harvard University found the first counterexample that proved Euler's conjecture was wrong. In other words, he found that the equation is true if a = 2,682,440, b = 15,365,639, c = 18,796,760 and d = 20,615,673. More recently, computer programmer Roger Frye of Thinking Machines Corp. in Cambridge, Mass., succeeded in finding the smallest positive integers that fit the equation. His exhaustive computer search showed that a = 95,800, b = 217,519, c = 414,560 and d = 422,560.

"I found my counterexample by a method that combined theoretical reasoning and a relatively short computer search," says Elkies. He converted the problem into an equivalent mathematical form that enabled him to pick out candidates likely to satisfy Euler's equation. A few other mathematicians had tried a similar approach, says Elkies, but either they gave up too soon or they were thinking in terms of proving rather than disproving the conjecture.

Once Elkies found the first counterexample, he was able to prove that there are infinitely many, each consisting of enormously large numbers. What Elkies didn't know yet was whether his initial solution was the smallest one.

Frye saw news of Elkies's achievement on a computer network bulletin board. Using hints supplied by Elkies to shorten the search, he wrote a computer program to look for smaller solutions. Working at night in his spare time, Frye used 110 hours of computer time on various Connection Machine computers before he was satisfied with the result.

No one knows whether another set of numbers, somewhere between those found by Frye and those discovered by Elkies, fits Euler's equation. "I've gone up to a million and not found a second set," says Frye. At that stage, the search begins to gobble up an excessive amount of computer time. Possible answers, although there are an infinite number of them, seem to be sparsely distributed.