Magazine article Science News

Prime Proof Zeros in on Crucial Numbers

Magazine article Science News

Prime Proof Zeros in on Crucial Numbers

Article excerpt

Fermat's last theorem is just one of many examples of innocent-looking problems that can long stymie even the most astute mathematicians. It took about 350 years to prove Fermat's tantalizing conjecture.

Now, Preda Mihailescu of the Swiss Federal Institute of Technology in Zurich has proved a theorem that is likely to lead to a solution of Catalan's conjecture, another venerable problem involving relationships among whole numbers. He describes his result in a paper to be published in the JOURNAL OF NUMBER THEORY.

"This is a very important contribution," says mathematician Andrew Granville of the University of Georgia in Athens. Mihailescu's work probably puts the resolution of Catalan's problem into the foreseeable future, he notes.

Named for Belgian mathematician Eugene Charles Catalan, the conjecture concerns powers of whole numbers. For example, the sequence of all squares and cubes of whole numbers greater than 1 begins with the integers 4, 8, 9, 16, 25, 27, and 36. In this sequence, 8 (the cube of 2) and 9 (the square of 3) are not only powers but also consecutive whole numbers.

In 1844, Catalan asserted that among powers of whole numbers, the only pair of consecutive numbers that arises is 8 and 9. Since then, Catalan's conjecture has posed a challenge to number theorists akin to that provided by Fermat's last theorem (SN: 11/5/94, p. 295).

Solving Catalan's problem amounts to a search for whole number solutions to the equation [x.sup.p] - [y.sup.q] = 1, where x, y, p, and q are all greater than 1. …

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