PreprintarXiv licencearXiv
On Fermat-type equations of signature (r,r,p)
Abstract
Let r≥11 be a prime. We show that there are infinitely many integers C for which the Fermat-type equation xr+yr=Czp 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| x+y, after which a further Frey curve and level lowering give a contradiction with the Ramanujan bound.
Related
- Entire solutions of Fermat-Type Partial Differential-Difference equations in higher dimensions
- On the a-number of Fermat type function fields and some of their subfields
- Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in ℂn
- The Fermat-Type Matrix Equation over GL2(ℤ)
- Signature Kernel and Schwinger-Dyson Kernel Equations as Two-Parameter Rough Differential Equations
BibTeX
@misc{freitas2026fermattype,
title = {{On Fermat-type equations of signature $(r,r,p)$}},
author = {Nuno Freitas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32509},
eprint = {2609.32509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32509},
}
RIS
TY - UNPB
TI - On Fermat-type equations of signature (r,r,p)
AU - Nuno Freitas
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32509
UR - https://arxiv.org/abs/2609.32509
AB - Let r≥11 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| x+y, after which a further Frey curve and level lowering give a contradiction with the Ramanujan bound.
ER -