Some remarks on the $abc$-conjecture

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 \[ \frac {{\log \max (|a|,|b|,|c|)}}{{\log r(abc)}}\] 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.