Abstract
AbstractLet K be a totally real field, and $$r\ge 5$$
r
≥
5
a fixed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions $$(x,y,z) \in \mathcal {O}_K^3$$
(
x
,
y
,
z
)
∈
O
K
3
of the signature (r, r, p) equation $$x^r+y^r=z^p$$
x
r
+
y
r
=
z
p
(where p is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subfields of $$K(\zeta _r)$$
K
(
ζ
r
)
. When $$K=\mathbb {Q}$$
K
=
Q
we get that for most of the primes $$r<150$$
r
<
150
with $$r \not \equiv 1 \mod 8$$
r
≢
1
mod
8
there are no non-trivial, primitive integer solutions (x, y, z) with 2|z for signatures (r, r, p) when p is sufficiently large. Similar results hold for quadratic fields, for example when $$K=\mathbb {Q}(\sqrt{2})$$
K
=
Q
(
2
)
there are no non-trivial, primitive solutions $$(x,y,z)\in \mathcal {O}_K^3$$
(
x
,
y
,
z
)
∈
O
K
3
with $$\sqrt{2}|z$$
2
|
z
for signatures (5, 5, p), (11, 11, p), (13, 13, p) and sufficiently large p.
Funder
Centre for Discrete Mathematics and its Applications, University of Warwick
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Cited by
1 articles.
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1. Two results on ^{}+^{}=^{};Proceedings of the American Mathematical Society;2023-11-07