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 logmax(lal, ibl, cil) log r(abc) equals 1. We show that in a natural anologue 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.