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be a positive nonsquare integer,
a prime number with
. We show that there exist effectively computable constants
such that if there is a solution to
, then for every
. As an application, we show that for
, if the equation
We solve the Diophantine equation
for all nonzero integers
. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.
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