The abc Conjecture and Square Free Parts of Fibonacci Numbers

In the paper [11], the second author considered a conjecture on the fundamental units of certain family of real quadratic fields related to Fibonacci numbers. In this paper, we shall investigate this conjecture more precisely in section 3, using the constant terms of the abc conjecture. We also prove the conjecture in section 4 for some special cases, using the integer points of several elliptic curves.