Some remarks on the 𝑎𝑏𝑐-conjecture

Abstract

Let r ( x ) 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 a + b + c = 0 , then the biggest limit point of the numbers \[ log ⁡ max ( | a | , | b | , | c | ) log ⁡ r ( a b c ) \frac {{\log \max (|a|,|b|,|c|)}}{{\log r(abc)}} \] equals 1. We show that in a natural anologue of this conjecture for n ≥ 3 n \geq 3 integers, the largest limit point should be replaced by at least 2 n − 5 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.