A note on Terai's conjecture concerning Pythagorean numbers (Q1281918)

Let \((a,b,c)\) be a primitive Pythagorean triple with \(2| a\). The authors prove that if \(b\equiv 1\pmod{16}\), \(b^2+1=2c,b\) and \(c\) are both odd primes, then the equation \(x^2+b^y=c^z\) has only the positive integer solution \((x,y,z)=(a,2,2)\).