Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Freitas, Nuno
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214715866