On Terai's conjecture concerning Pythagorean numbers (Q2759143)

A conjecture of Terai is as follows: Let \((a,b,c)\) be a primitive Pythagorean triple such that \(a^2+ b^2= c^2\) where \(a,b,c\) are natural numbers with \(\gcd (a,b,c)= 1\) and \(a\) even. Then the equation \(x^2+ b^y= c^z\) in natural integers \(x,y,z\) has only the solution \((x,y,z)= (a,2,2)\). The author proved the conjecture [Acta Arith. 71, 253-257 (1995; Zbl 0820.11023)] if \(b> 8\cdot 10^6\), \(b\equiv \pm 5\pmod 8\) and \(c\) is a prime power. Later \textit{Z. Cao} and \textit{X. Dong} [Proc. Japan Acad., Ser. A 74, 127-129 (1998; Zbl 0924.11024)] and \textit{P. Yuan} and \textit{J. Wang} [Acta Arith. 84, 145-147 (1998; Zbl 0895.11016)] proved the conjecture if \(b\equiv \pm 5\pmod 8\) and either \(b\) or \(c\) is a prime. In this paper, the author proves the conjecture if \(b\equiv 7\pmod 8\) and either \(b\) is a prime or \(c\) is a prime power.