On Terai's conjecture

抄録

Let p be an odd prime such that b^r + 1 = 2p^t, where r, t are positive integers and b ≡ 3,5 (mod 8). We show that the Diophantine equation x² + b^m = pⁿ has only the positive integer solution (x,m,n) = (p^t-1,r,2t). We also prove that if b is a prime and r = t = 2, then the above equation has only one solution for the case b ≡ 3,5,7 (mod 8) and the case d is not an odd integer greater than 1 if b ≡ 1 (mod 8), where d is the order of prime divisor of ideal (p) in the ideal class group of Q ().