Solaleh Bolvardizadeh (University of Lethbridge, Canada)
The quality of the triplet (a,b,c), where gcd, satisfying a + b = c is defined as
q(a,b,c) = \frac{\max\{\log |a|, \log |b|, \log |c|\}}{\log \mathrm{rad}(|abc|)},
where \mathrm{rad}(|abc|) is the product of distinct prime factors of |abc|. We call such a triplet an ABC-solution. The ABC-conjecture states that given \epsilon > 0 the number of the ABC-solutions (a,b,c) with q(a,b,c) \geq 1 + \epsilon is finite.
In the first part of this talk, under the ABC-conjecture, we explore the quality of certain families of the ABC-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of ABC-solutions that has quality > 1.
In the remaining of the talk, we prove a conjecture of Erd\"os on the solutions of the Brocard-Ramanujan equation
n! + 1 = m^2
by assuming an explicit version of the ABC-conjecture proposed by Baker.