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Problem of the Week 952

ABC Triples

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An "abc triple" is a set of three positive integers a, b, c such that a + b = c, a and b have no common factor, and r, the product of all primes that divide a, b, or c, is less than c. The "quality" of the triple is the ratio log c/log r. Find an abc-triple with high quality.

Example: 1 + 8 = 9 is an abc triple because r is 6 which is less than 9. Its quality is log 9/log 6, or 1.23. 3+125=128, with quality 1.42 is another example.

Source: Hendrik Lenstra's fabulous lecture at the math meetings in San Diego, Jan 2002. The famous "abc conjecture" by Masser & Osterle is the assertion that the quality has the limit 1. This conjecture is amazingly strong and implies Fermat's Last Theorem, Catalan's Conjecture, and numerous other proved and unproved results of number theory.

For the latest information on ABC Triples, see http://www.math.leidenuniv.nl/~desmit/abc/

© Copyright 2002 Stan Wagon. Reproduced with permission.

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12 February 2002