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January 2001
Race to settle Catalan conjecture: it's people vs. computers. Ivars Peterson reports in the December 2, 2000 Science News on recent progress towards the resolution of this 150yearold conjecture. Catalan noted that 8=2^{3} and 9=3^{2} are consecutive integers and conjectured that they were the only set of consecutive whole powers. This translates to the Fermatlike statement that the equation x^{p}y^{q}=1 has no wholenumber solutions other than 3^{2}  2^{3} = 1. Recently Maurice Mignotte (Strasbourg) had given upper bounds on possible values of p and q; now Preda Mihailescu (ETH, Zürich) has shown that p and q must be a pair of ``double Weiferich primes." Only six pairs are known, and, as Peterson reports, ``a major collaborative computational effort has been mounted" to find more. You can help: volunteer at Ensor Computing's Catalan Conjecture page. Or you can join mathematicians who ``are betting that a theoretical approach will beat out the computers." Prez mentions math. Bill Clinton gave a long interview to Science editor Ellis Rubinstein (printed in the December 22, 2000 issue). He spoke warmly and knowledgeably about the scientific challenges facing the US and the world in the near future. He did not mention the Catalan Conjecture, but he did dwell at some length on the need to improve science literacy in the population. ``... I think the language of science and the necessity of understanding at least the basic concepts of science will become a much more pervasive part of the average citizen's life in the next 20 to 30 years than it ever has been." He addressed the education problem directly: ``... I think there are basically two issues. One is, in a country as big and diverse as ours, how do you get more kids to take math and science courses at more advanced levels? And secondly, if you could do that, how would you have enough qualified teachers to do it?" He mentioned several approaches to the second problem, including ``find ways to finance the education of young people to do this work for 4 or 5 years" and ``have in the junior and senior high schools people who have this knowledge and yet who come in and teach a single course ..." The full text of the interview is available online. "Rhetoric and the Math Melodrama" is the title of David Foster Wallace's review of two recent mathbased novels in the December 22 2000 Science. This was an inspired choice of reviewer. Wallace rose to the occasion with a wonderful, 24footnote essay on mathematics and the people who make it. ``Math's cultural stock has risen hard in recent years," it begins, and goes on to survey the phenomenon, from Good Will Hunting to the iconification of Andrew Wiles. The books in question, ``The Wild Numbers" by Philibert Schogt (Four Walls Eight Windows, New York) and ``Uncle Petros & Goldbach's Conjecture" (Bloomsbury USA, New York) prompt his statement of what we may call Wallace's Paradox: ``The type of audience most likely to accept and appreciate these novels' lofty, encomiastic view of pure math is also the audience most apt to be disappointed by the variously vague, reductive or inconsistent ways the novels handle the actual mathematics they're concerned with." Nevertheless he does make one want to read the second. ``...the embedded story of Petros's fall is in fact a kind of monstrous gem, one in whose facets readers of many different backgrounds and tastes may see parts of themselves reflected." "Beautiful Minds" is reportedly the title of a movie to be directed by Ron Howard (How the Grinch Stole Christmas), starring Russell Crowe, Ed Harris, and Jennifer Connolly. "It's based on a true story about a brilliant mathematician named John Nash, who ultimately won the Nobel Prize but had to overcome a tremendous amount of adversity and led a fascinating life," Howard told Steven Silverman, who interviewed him for People.com The Daily, datelined Dec 11 2000. Howard went on: "It's a great story with great roles for actors. Russell Crowe will be extraordinary in it." Method and madness. Carlin Romano reviewed two biographies of mathematicians on the September 8, 2000 Chronicle of Higher Education. The books, "The mystery of the Aleph: Mathematics, the Kaballah and the Search for Infinity" by Amir Aczel (on Georg Cantor; Four Walls, Eight Windows) and "Gödel: A Life of Logic" by John Casti and Werner DePauli (Perseus Books), both deal with great mathematicians who were tormented by mental disease. Romano quotes Seneca: "There is no great genius without some touch of madness," but these two men's afflictions went well beyond "a touch." Russell's paradox apparently gets referred to in both books, and Romano speculates, as I understand it, on whether the barber who shaves everyone who does not shave himself could have a parallel in these scientists, who could think for humanity but somehow could not reason their own way to a peaceful existence. Tony Phillips 
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