Let K be a totally real number field and consider a Fermat‐type equation A ap+ B bq= C cr
Aa^p+Bb^q=Cc^r over K. We call the triple of exponents (p, q, r) (p,q,r) the signature of the …

In the first part we construct algorithms (over $\mathbb {Q} $) which we apply to solve $ S $-
unit, Mordell, cubic Thue, cubic Thue–Mahler and generalized Ramanujan–Nagell …

Let K be a totally real field, and 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 …

Let $ K $ be a totally real number field with narrow class number one and $ O_K $ be its ring
of integers. We prove that there is a constant $ B_K $ depending only on $ K $ such that for …

Abstract The $ S $-unit equation $\alpha x+\beta y= 1$ in $ x, y\in\mathcal O_S^\times $
plays a very important role in Diophantine number theory. We first present the best known …

Recent results of Freitas, Kraus, Sengun and Siksek give sufficient criteria for the asymptotic
Fermat's Last Theorem to hold over various number fields. In this paper, we prove …

Given a smooth, proper, geometrically integral curve $ X $ of genus $ g $ with Jacobian $ J
$ over a number field $ K $, Chabauty's method is a $ p $-adic technique to bound $\# X (K) …

Let $ K $ be a totally real number field, and $\mathcal {O} _K $ be the ring of integers of $ K
$. In this article, we study the asymptotic solutions of the generalized Fermat equation, ie …

The unit equation over cyclic number fields of prime degree Page 247 msp ALGEBRA AND
NUMBER THEORY 15: 10 (2021) https://doi. org/10.2140/ant. 2021.15. 2647 The unit equation …

Andrews and Petsche proposed in 2020 a conjectural characterization of all pairs $(f,\alpha)
$, where $ f $ is a polynomial over a number field $ K $ and $\alpha\in K $, such that the …