Prime effort: powerful conjecture may be proved. (This Week).Innocent-looking problems involving whole numbers can stymie sty·mie also sty·my tr.v. sty·mied , sty·mie·ing also sty·my·ing , sty·mies To thwart; stump: a problem in thermodynamics that stymied half the class. n. 1. even the most astute mathematicians. As in the case of Fermat's last theorem Fermat's last theorem Statement that there are no natural numbers x, y, and z such that xn + yn = zn, in which n is a natural number greater than 2. , centuries of effort may go into proving such tantalizing tan·ta·lize tr.v. tan·ta·lized, tan·ta·liz·ing, tan·ta·liz·es To excite (another) by exposing something desirable while keeping it out of reach. , deceptively simple conjectures in number theory (SN: 11/5/94, p. 295). Now, Preda Mihailescu of the University of Paderborn The University of Paderborn (German: Universität Paderborn) in Paderborn, North Rhine-Westphalia, Germany was founded in 1972. 14,700 students were enrolled at the university as of October 2005. in Germany finally may have the key to a venerable problem known as Catalans conjecture, which concerns powers of whole numbers. Consider the sequence of all squares and cubes of whole numbers greater than 1, a sequence that 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, Belgian mathematician Eugene Charles Catalan asserted that, among all powers of whole numbers, the only pair of consecutive integers is 8 and 9. 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. The conjecture proposes that there is only one such solution: [3.sup.2] - [2.sup.3] = 1. A breakthrough in solving the problem occurred in 1976 when Robert Tijdeman of the University of Leiden in the Netherlands showed that there is a finite rather than an infinite number infinite number a number so large as to be uncountable. Represented by 8, frequently obtained by 'dividing' by zero. of solutions to the equation. In 2000, Mihailescu proved that if additional solutions to the equation exist, the pair of exponents must be of a rare type known as double Wieferich primes (SN: 12/2/00, p. 357). A prime is a whole number evenly divisible DIVISIBLE. The susceptibility of being divided. 2. A contract cannot, in general, be divided in such a manner that an action may be brought, or a right accrue, on a part of it. 2 Penna. R. 454. only by itself and 1. Mihailescu continued to work on the problem, and he apparently cracked it earlier this year. He has now sent a draft of his purported proof of Catalan's conjecture to several mathematicians for checking. It isn't certain yet that Mihailescu's proof will hold up, but there are encouraging signs. Yuri F. Bilu of the University of Bordeaux University of Bordeaux can refer to one or all of the four universities in Bordeaux, each of which covers a different field of study:
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