Exceptional cases of Terai's conjecture on Diophantine equations

Let p,q,r be positive integers with p,q,r greater than or equal to 2, and let a,b,c be relatively prime positive integers with a(P) + b(q) = c(r). Terai conjectured that (apart from a handful of known exceptions) the only solution of the equation a(x) + b(y) = c(z) in positive integers x,y,z is (x,y,z) = (p,q,r). In this article, we consider the case q = r = 2 and give some results related to exceptional cases.