• U.S.

# Science: Eureka!

TIME

In the third century B.C., a Greek mathematician named Archimedes jumped from his bath, rushed home naked and dripping, shouting “Eureka, eureka!” He had just discovered an important physical principle. In 1937 A.D., a German-Jewish mathematician named Samuel Isaac Krieger, who was taking a mineral bath near Buffalo, N. Y., suddenly leaped out, rushed naked into the adjoining room, began to scribble figures. He thought he had discovered something too: a solution to the equation given in Fermat’s last theorem.

The last theorem of French Mathematician Pierre Fermat, laid down in the 17th Century states that there are no solutions to the equation: x<SUP>n</SUP>+y<SUP>n</SUP> = z<SUP>n</SUP>, n being a power greater than the square and x, y and z being whole numbers which are not zero.* Fermat wrote on the margin of a book that he had hit upon a proof of the theorem, but that there was not room enough on the margin to write it out. He died before he wrote it anywhere else that anyone knew of. The theorem became celebrated in the history of mathematics, and dozens of able number-jugglers tried to prove it. Their efforts led to valuable additions to mathematical theory, and they did prove that there are no solutions to the Fermat equation in certain cases, with values of n up to 7,000. But they did not prove that there are no solutions in all cases, with any value of n. At the Uni versity of Gottingen in 1907 the Wolfs-kehl prize of 100,000 marks was established, to reward anyone who could offer such proof.

Samuel Isaac Krieger arrived in the U. S. from Germany some ten years ago, self-billed as a mathematical wizard and armed with a letter purporting to be a yip of praise from no less a personage than Albert Einstein. He quickly convinced reporters that he was indeed a marvel at quick mental calculation. He would say, “Think of a number from one to a bil lion,” multiply the number given by a smaller number and have the answer in a few seconds. He would ask a newshawk for the date of his birth and then, after a moment of cogitation, tell him what day of the week it had been.

When he hit on his bathroom solution of Fermat’s equation, Krieger at once cabled to Göttingen asking whether the 100,000-mark prize was still there. Back came the answer: “Preis besteht noch” (Prize still stands). Krieger doubted, however, that Adolf Hitler would allow the money to leave Germany, especially since the claimant was conspicuously non-Aryan. A matter which he apparently overlooked was that the prize is offered for proof of the theorem, whereas his solution, if valid, would constitute disproof.

Last week Herr Krieger made headlines once more by announcing that he would reveal the values for x, y and z which would solve the Fermat equation. They turned out to be 1,324; 731; and 1,961. He would not reveal n—the power—but said it was less than 20. An astute reporter from the New York Times, no baby in mathematics himself, pored over this equation: 1,324<SUP>n</SUP>+731<SUP>n</SUP>=1,961<SUP>n</SUP>. The reporter saw that the first number raised to any power at all would end in either 6 or 4, the second raised to any power would end in 1, and the third raised to any power would end in 1. Therefore Herr Krieger was making the astonishing assertion that a number ending in 6 or 4, added to a number ending in 1, would give a total ending in 1. The Times man got Krieger on the telephone, explained this.

“You mean,” said Krieger, “that you doubt me?” The reporter admitted it.

“Well,” said Samuel Isaac Krieger sadly, “when the time comes, I will explain everything.”

*If n is the second power or square, there are many solutions. E. g., 3<SUP>2</SUP>+4<SUP>2</SUP> = 5<SUP>2</SUP> (9 + 16=25).