Exceptional cases of Terai’s conjecture on Diophantine equations

Let p,q,r be positive integers with $p,q,r≥2,$ and let a,b,c be relatively prime positive integers with $ap+bq = cr.$ Terai conjectured that (apart from a handful of known exceptions) the only solution of the equation $ax+by = cz$ 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.