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Some Remarks on the abc-Conjecture

J. Browkin and J. Brzeziński
Mathematics of Computation
Vol. 62, No. 206 (Apr., 1994), pp. 931-939
DOI: 10.2307/2153551
Stable URL: http://www.jstor.org/stable/2153551
Page Count: 9
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Some Remarks on the abc-Conjecture
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Abstract

Let r(x) be the product of all distinct primes dividing a nonzero integer x. The abc-conjecture says that if a, b, c are nonzero relatively prime integers such that a + b + c = 0, then the biggest limit point of the numbers log max(|a|, |b|, |c|)/log r(abc) equals 1. We show that in a natural analogue of this conjecture for n ≥ 3 integers, the largest limit point should be replaced by at least 2n - 5. We present an algorithm leading to numerous examples of triples a, b, c for which the above quotients strongly deviate from the conjectural value 1.

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