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Alejandro Argáez-García

Publications and source records attributed to Alejandro Argáez-García.

6 recordsLinked to original sources
arXiv2025

On the solutions to $Ax^p+By^p+Cz^p=0$ over quadratic fields

We provide the necessary conditions for the existence of solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ over any quadratic number field $K$ with $A,B,C$ pth powerfree integer numbers. We determine when $x$, $y$ and $z$ are rational numbers for pairwise coprime integers $A$, $B$ and $C$. Moreover, we prove that $x$, $y$ and $z$ are in $K\setminus\mathbb{Q}$ when $BC=\pm 1$ and $A\neq \pm 2$. Finally, we prove that no solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ exist in $K\setminus\mathbb{Q}$ when $BC\neq \pm 1$.

math.NT↗
arXiv2025

Computing rational solutions to $Ax^p+By^p+Cz^p=0$

In this survey, we studied the possibility of finding rational solutions to the equation $Ax^p+By^p+Cz^p=0$ via its attached hyperelliptic curve $Y^2=X^p+A^2(BC)^{p-1}/4$ and its rational points computed using computational tools.

math.NT↗
arXiv2019

On perfect powers that are sums of cubes of a seven term arithmetic progression

We prove that the equation $(x-3r)^3+(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3+(x+3r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq r\leq 10^6$ and $p\geq 5$ prime. This article complements the work on the equations $(x-r)^3 + x^3 + (x+r)^3 = y^p$ and $(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3= y^p$ . The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.

math.NT↗
arXiv2018

Black Box Galois Representations

We develop methods to study $2$-dimensional $2$-adic Galois representations $ρ$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on $K$ and $S$, we show how to determine the determinant $\detρ$, whether or not $ρ$ is residually reducible, and further information about the size of the isogeny graph of $ρ$ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for $K=\mathbb{Q}$, and for $K$ imaginary quadratic, $ρ$ being the representation attached to a Bianchi modular form. These results form part of the first author's thesis.

math.NT↗
arXiv2017

On perfect powers that are sums of cubes of a three term arithmetic progression

Using only elementary arguments, Cassels and Uchiyama (independently) determined all squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result and determined all perfect powers that are sums of three consecutive cubes. Recently, the equation $(x-r)^k + x^k + (x+r)^k$ has been studied for $k=4$ by Zhongfeng Zhang and for $k=2$ by Koutsianas. In this paper, we complement the work of Cassels, Koutsianas and Zhang by considering the case when $k=3$ and showing that the equation $(x-r)^3+x^3+(x+r)^3=y^n$ with $n\geq 5$ a prime and $0 < r \leq 10^6$ only has trivial solutions $(x,y,n)$ which satisfy $xy=0$.

math.NT↗