On the system of Diophantine equations \(a^2+b^2=(m^2+1)^r\) and \(a^x+b^y=(m^2+1)^z\)
Let \(a, b, m, r\) be positive integers such that \(a^2+ b^2= (m^2+ 1)^r\), \(\gcd(a,b)= 1\), \(2\mid m\), \(2\nmid r\) and \(r> 1\). By a wide conjecture proposed by \textit{N. Terai} [Proc. Japan Acad., Ser. A 70, No. 1, 22--26 (1994; Zbl 0812.11024)], the equation NEWLINE\[NEWLINEa^x+ b^y= (m^2+ 1)^z\tag{\(*\)}NEWLINE\]NEWLINE should have only the positive integer solution \((x,y,z)= (2,2,r)\). There have been many papers related to this problem. For example, the reviewer [Acta Math. Sin., Chin. Ser. 53, No. 6, 1239--1248 (2010; Zbl 1240.11060)] proved that if \(m> 10^6 r^6\), then \((*)\) has only the solution \((x,y,z)= (2,2,r)\).NEWLINENEWLINE In this paper, using a synthetical analysis, the author proves that \((*)\) admits a solution \((x,y,z)\neq (2,2,r)\) only in finitely many instances \((m,r)\), and all such solutions are computable.
scientific article; zbMATH DE number 6033773