  Let (a,b,c) be a primitive Pythagorean triple, i.e., a^{2}+b^{2}=c^{2} with
gcd(a,b,c)=1, a even and b odd. Terai's conjecture says that the Diophantine
equation x^{2}+b^{y}=c^{z} has only the positive integer solutions
(x,y,z)=(a,2,2). In this study, we prove that Terai's conjecture is true when b
is a product of two primes and c=5(mod8).
