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In this paper, we are interested in solving the Fermat-type equations x 5 + y 5 = dz P , where d is a positive integer and p a prime number > 7. We describe a new method based on modularity theorems which allows us to improve all earlier results for this equation. We finally discuss the present limits of the method by looking at the case d = 3.

Solving Fermat-type equations $x^5+y^5=dz^p$