Angelos Koutsianas

activity

20152026

most citedConductor and discriminant of Picard curves

12 citations · 17 across the 10 of their papers we have counts for

collaborators

15 papers

math.NT2026

Sum of consecutive powers as a perfect power

Angelos Koutsianas, Nikos Tzanakis

In this paper we study the equation $x^{k} + (x + 1)^{k} = y^{n}, n \geq 3,$ when $k \equiv 2 (mod 4)$ . We prove that the only solutions are for $x = 0, - 1$ when $6 \leq k \leq 100$ or…

math.NT2026

The generalized Fermat equation $A x^{2} + B y^{r} = C z^{p}$ and applications

Pedro-José Cazorla García, Angelos Koutsianas, Lucas Villagra-Torcomian

In this paper, we develop the modular method for the generalized Fermat equation appearing in the title, within the framework of Darmon's program and using Frey hyperelliptic curve…

math.NT2025

Darmon's Program: A survey

Imin Chen, Angelos Koutsianas

We give an overview of Darmon's program for resolving families of generalized Fermat equations with one varying exponent and survey what is currently known about this approach base…

math.NT2025

Local-global principle for 11-isogenies of elliptic curves is true over quadratic fields

Stevan Gajović, Jeroen Hanselman, Angelos Koutsianas

In this paper, we prove that the local-global principle of $11$ -isogenies for elliptic curves over quadratic fields does not fail. This gives a positive answer to a conjecture by B…

math.NT2022★ 3 cited

A modular approach to Fermat equations of signature $(p, p, 5)$ using Frey hyperelliptic curves

Imin Chen, Angelos Koutsianas

In this paper we carry out the steps of Darmon's program for the generalized Fermat equation $x^{n} + y^{n} = z^{5} .$ In particular, we develop the machinery necessary to prove an op…

math.AG2021★ 2 cited

Weak Approximation for $0$ -cycles on a product of elliptic curves

Evangelia Gazaki, Angelos Koutsianas

In the 1980's Colliot-Thélène, Sansuc, Kato and S. Saito proposed conjectures related to local-to-global principles for $0$ -cycles on arbitrary smooth projective varieties over a n…

math.NT2020

Perfect powers in sum of three fifth powers

Pranabesh Das, Pallab Kanti Dey, Angelos Koutsianas +1

In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation $$ (x-d…

math.NT2020

The equation $(x - d)^{5} + x^{5} + (x + d)^{5} = y^{n}$

Michael A. Bennett, Angelos Koutsianas

In this paper, we solve the equation of the title under the assumption that $g cd (x, d) = 1$ and $n \geq 2$ . This generalizes earlier work of the first author, Patel and Siksek [BPS16]…

math.NT2019

A robust implementation for solving the $S$ -unit equation and several applications

Alejandra Alvarado, Angelos Koutsianas, Beth Malmskog +3

Let $K$ be a number field, and $S$ a finite set of places in $K$ containing all infinite places. We present an implementation for solving the $S$ -unit equation $x + y = 1$ , $x,y \i…

math.NT2019★ 12 cited

Conductor and discriminant of Picard curves

Irene I. Bouw, Angelos Koutsianas, Jeroen Sijsling +1

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $Q$ wit…

math.NT2018

An application of the modular method and the symplectic argument to a Lebesgue-Nagell equation

Angelos Koutsianas

In this paper, we study the generalized Lebesgue-Nagell equation \[ x^2+7^{2k+1}=y^n. \] This is the last case of equations of the form $x^{2} + q^{2 k + 1} = y^{n}$ with $k \geq 0$ and $q > 0$ w…

math.NT2018

On the generalized Fermat equation $a^{2} + 3 b^{6} = c^{n}$

Angelos Koutsianas

In this paper, we prove that the only primitive solutions of the equation $a^{2} + 3 b^{6} = c^{n}$ for $n \geq 3$ are $(a, b, c, n) = (\pm 47, \pm 2, \pm 7, 4)$ . Our proof is based on the modularity…

math.NT2018

Perfect powers that are sums of squares in a three term arithmetic progression

Angelos Koutsianas, Vandita Patel

We determine primitive solutions to the equation $(x - r)^{2} + x^{2} + (x + r)^{2} = y^{n}$ for $1 \leq r \leq 5, 000$ , making use of a factorization argument and the Primitive Divisors Theorem…

math.NT2017

On the solutions of the Diophantine equation $(x - d)^{2} + x^{2} + (x + d)^{2} = y^{n}$ for $d$ a prime power

Angelos Koutsianas

In this paper, we determine the primitive solutions of the Diophantine equation $(x - d)^{2} + x^{2} + (x + d)^{2} = y^{n}$ when $n \geq 2$ and $d = p^{b}$ , $p$ a prime and $p \leq 1 0^{4}$ . The main ingredi…

math.NT2015

Computing all elliptic curves over an arbitrary number field with prescribed primes of bad reduction

Angelos Koutsianas

In this paper we study the problem of how to determine all elliptic curves defined over an arbitrary number field $K$ with good reduction outside a given finite set of primes $S$ o…

Sum of consecutive powers as a perfect power

The generalized Fermat equation Ax2+Byr=Czp and applications

Darmon's Program: A survey

Local-global principle for 11-isogenies of elliptic curves is true over quadratic fields

A modular approach to Fermat equations of signature (p,p,5) using Frey hyperelliptic curves

Weak Approximation for 0-cycles on a product of elliptic curves

Perfect powers in sum of three fifth powers

The equation (x−d)5+x5+(x+d)5=yn

A robust implementation for solving the S-unit equation and several applications

Conductor and discriminant of Picard curves

An application of the modular method and the symplectic argument to a Lebesgue-Nagell equation

On the generalized Fermat equation a2+3b6=cn

Perfect powers that are sums of squares in a three term arithmetic progression

On the solutions of the Diophantine equation (x−d)2+x2+(x+d)2=yn for d a prime power

Computing all elliptic curves over an arbitrary number field with prescribed primes of bad reduction

The generalized Fermat equation $A x^{2} + B y^{r} = C z^{p}$ and applications

A modular approach to Fermat equations of signature $(p, p, 5)$ using Frey hyperelliptic curves

Weak Approximation for $0$ -cycles on a product of elliptic curves

The equation $(x - d)^{5} + x^{5} + (x + d)^{5} = y^{n}$

A robust implementation for solving the $S$ -unit equation and several applications

On the generalized Fermat equation $a^{2} + 3 b^{6} = c^{n}$

On the solutions of the Diophantine equation $(x - d)^{2} + x^{2} + (x + d)^{2} = y^{n}$ for $d$ a prime power