We study the Diophantine equation
formula here
in integers x, y > 1, n > 2 and q [ges ] 2.
Without loss of generality, we suppose that q is prime.
Saradha and Shorey [SS] considered equation (1·1) with
x = z2 and proved that it has no solutions in the cases
z > 31 or z ∈ {2, 3, 4, 8, 9, 16, 27}.Notice that the proof for z ∈ {2, 4, 8, 16} is elementary,
whereas for z ∈ {3, 9, 27} the proof uses a result of Darmon–Merel
[DM] which is a generalization of Wiles' result on Fermat's
theorem. The purpose of the present paper is to treat completely the remaining cases
and to prove the following result.