The solutions of a Diophantine equation (Q2742201)

Using some results of the reviewer [Kexue Tongbao 32, 1519-1521 (1987; Zbl 0634.10018)], \textit{M. Toyoizumi} [Comment. Math. Univ. St. Paul. 27, 105-111 (1978; Zbl 0421.10013)] and \textit{F. Beukers} [Acta Arith. 38, 389-410 (1981; Zbl 0454.10009)], the authors prove by elementary methods that the Diophantine equations \(103^x+b^y=2^z\), where \(b\) is prime, \(3\leq b<200\), have no solutions \(x,y,z\) in positive integers except \(103+5^2=2^7\).NEWLINENEWLINENEWLINE\{Reviewer's remark: Let \(a,b\) and \(c\) be distinct primes. Nagell, the reviewer, et al. [see \textit{Z. Cao}, Introduction to Diophantine equations (Chinese), Haerbin Gongye Daxue Chubanshe, Harbin (1989; Zbl 0849.11029)] gave all positive integer solutions of the Diophantine equations (1) \(a^x+b^y=c^z,\;\max(a,b,c)<100\). In 1988 and 1991, the reviewer [Kexue Tongbao 33, No. 3, 237 (1988); Ziran Zazhi 14, No. 11, 872-873 (1991)] also gave all positive integer solutions of (1) when \(100<\max(a,b,c)<200\), and proved that if \(\max(a,b,c)>13\), then (1) has at most one solution in positive integers \(x,y,z\) with \(z>1\).\}.