ON A LIMIT POINT ASSOCIATED WITH THE abc{CONJECTURE

Let Q(n) denote the squarefree part of n so that Q(n) = Q pjn p. Throughout, we set a,b, and c to be positive relatively prime integers with c = a + b. Deene L a;b = log c log Q(abc) : The abc-conjecture of Masser and Oesterl e asserts that the greatest limit point of the double sequence fL a;b g is 1. Recently, in joint work with Browkin, Greaves, Schinzel, and the rst author 1], it was shown that the abc-conjecture is equivalent to the assertion that the precise set S of limit points of fL a;b g is the interval 1=3; 1]. Unconditionally, using certain polynomial identities and a theorem concerning squarefree values of binary forms, they showed that 1=3; 15=16] S. Further polynomial identities of Greaves and Nitaj (private communication) imply that 1=3; 36=37] S. By considering a = 1 and b = 2 n , it is easy to see that fL a;b g has a limit point 1 in the extended real line. The purpose of this note is to establish the following: In other words, we prove that there is a limit point of fL a;b g somewhere in the interval 1; 3=2). Before proving the theorem, it is of some value to discuss simpler arguments for two weaker results. First, we observe that the existence of a nite limit point 1 can be established as follows. Fix a positive integer k, and let n 2 be a squarefree number. Observe that n Q(n(n k ? 1)) n k+1 :