An application of Frey's idea to exponential diophantine equations (Q1344794)

Let \(a\), \(b\), \(c\), \(l\), \(m\), \(n\) be relatively prime positive integers. In this paper it is shown that the equation \(la^ x+ mb^ y= nc^ z\), has a finite number of solutions in positive integers \(x\), \(y\), \(z\), all of which can be effectively determined. The effective procedure is based on: a) \textit{G. Frey}'s method which reduces Fermat's problem to the existence of a certain kind of elliptic curve [Ann. Univ. Sarav., Ser. Math. 1, 1-40 (1986; Zbl 0586.10010)], b) Shafarevich-Coates theorem which gives an effective method to determine (up to isomorphism) all the elliptic curves over \(\mathbb{Q}\) having good reduction at all primes outside of a fixed finite set of prime numbers [\textit{J. Coates}, Acta Arith. 16, 425-435 (1970; Zbl 0221.10027)].