On Fermat-type equations of signature $(r,r,p)$

Abstract

Let $r\geq11$ be a prime. We show that there are infinitely many integers $C$ for which the Fermat-type equation $$ x^r+y^r=Cz^p $$ has no non-trivial primitive solutions for all sufficiently large (in terms of $r$ and~$C$) prime exponents~$p$. The proof uses several Frey curves to force simultaneous Frobenius trace equalities at the primes above~$3$; a new trace separation argument shows that these equalities imply $3\mid x+y$, after which a further Frey curve and level lowering give a contradiction with the Ramanujan bound.