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Ivars Peterson's MathTrek |
June 24, 2002
Innocent-looking problems involving whole numbers can stymie even the most astute mathematicians. As in the case of Fermat's last theorem, centuries of effort may go into proving such tantalizing, deceptively simple conjectures in number theory.
Now, Preda Mihailescu of the University of Paderborn in Germany finally may have the key to a venerable problem known as Catalan's conjecture, which concerns the 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 of integers but also consecutive whole numbers.
In 1844, Belgian mathematician Eugène Charles Catalan (1814–1894) 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 xp – yq = 1, where x, y, p, and q are all greater than 1. The conjecture proposes that there is only one such solution: 32 – 23 = 1.
Interestingly, more than 500 years before Catalan formulated his conjecture, Levi ben Gerson (1288–1344) had already shown that the only powers of 2 and 3 that differ by 1 are 32 and 23.
A breakthrough in solving the problem occurred in 1976 when Robert Tijdeman of the University of Leiden in the Netherlands shows that, should the conjecture not hold, there can be only a finite rather than an infinite number of solutions to the equation. In effect, each of the exponents p and q must be less than a certain value, initially shown to be astronomically huge but later reduced to more manageable levels.
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. These prime numbers obey the following relationship: p(q – 1) must leave a remainder of 1 when divided by q2, and q(p – 1) must leave a remainder of 1 when divided by p2.
Double Wieferich primes are extremely rare. Only six examples have been identified so far: 2 and 1,093; 3 and 1,006,003; 5 and 1,645,333,507; 83 and 4,871; 911 and 318,917; and 2,903 and 18,787. None of these pairs are relevant to the question of proving Catalan's conjecture.
Mihailescu continued to work on the problem, and he apparently cracked it earlier this year. His proof of Catalan's conjecture takes advantage of his earlier result on double Wieferich primes, Mihailescu says.
It isn't absolutely certain yet that Mihailescu's proof will hold up, but there are very encouraging signs. Yuri F. Bilu of the University of Bordeaux I in Talence, France, has analyzed Mihailescu's work and written a favorable commentary outlining the proof's main steps. "I am sure that Mihailescu's proof is correct," Bilu declares.
Mihailescu presented the proof publicly for the first time on May 24 at a Canadian Number Theory Association meeting in Montreal. His presentation was well received and, encouraged by the positive response from several prominent number theorists, Mihailescu is now preparing a manuscript of his proof for publication.
It looks like Catalan's conjecture is about to join the mathematical pantheon of illustrious theorems.
Copyright 2002 by Ivars Peterson
References:
Bilu, Y.F. Preprint. Catalan's conjecture (after Mihailescu). See http://www.ufr-mi.u-bordeaux.fr/~yuri/.
Granville, A. 2001. The latest on Catalan's conjecture. MAA Focus (May/June):4-5.
Mihailescu, P. Preprint. Primary cyclotomic units and a proof of Catalan's conjecture.
______. In press. A class number free criterion for Catalan's conjecture. Journal of Number Theory
Peterson, I. 2002. Prime effort: Powerful conjecture may be proved. Science News 161(May 25):324.
______. 2000. Prime proof zeros in on crucial numbers. Science News 158(Dec. 2):357.
______. 2000. Zeroing in on Catalan's conjecture. MAA Online (Dec. 4).
103(August-September):529-538.
Additional information about Catalan's conjecture is available at http://mathworld.wolfram.com/CatalansConjecture.html and http://primes.utm.edu/glossary/page.php/CatalansProblem.html.
A definition of a double Wieferich prime pair can be found at http://mathworld.wolfram.com/DoubleWieferichPrimePair.html.
A brief biography of Eugène Charles Catalan can be found at http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Catalan.html.
Information about the Canadian Number Theory Association meeting in Montreal can be found at http://www.math.mcgill.ca/cnta7/.
Preda Mihailescu has a Web page at http://www-math.uni-paderborn.de/~preda/.
Comments are welcome. Please send messages to Ivars Peterson at ip@sciserv.org.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles.