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Mathematics of Computation

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Some remarks on the $ abc$-conjecture


Authors: J. Browkin and J. Brzeziński
Journal: Math. Comp. 62 (1994), 931-939
MSC: Primary 11D04; Secondary 11A55, 11Y65
DOI: https://doi.org/10.1090/S0025-5718-1994-1218341-2
MathSciNet review: 1218341
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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

$\displaystyle \frac{{\log \max (\vert a\vert,\vert b\vert,\vert c\vert)}}{{\log r(abc)}}$

equals 1. We show that in a natural anologue of this conjecture for $ n \geq 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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DOI: https://doi.org/10.1090/S0025-5718-1994-1218341-2
Article copyright: © Copyright 1994 American Mathematical Society