The abc-conjecture is true for at least $N(c),1\le N(c)<\varphi (c)/2$, partitions a, b of c

Abstract

We prove that for any positive integer c there are at least N(c), $1\le N(c)<\varphi (c)/2$ representations of c as a sum of two positive integers a, b, with no common divisor, such that the N(c) radicals R(abc) are all greater than kc, where k an absolute constant.

9 pages