Tobias Hilgart

collaborators

3 papers

math.NT2024

On the unique solvability of the simultaneous Pell equations $x^{2} - a y^{2} = 1$ and $z^{2} - b x^{2} = 1$

Tobias Hilgart, Volker Ziegler

We consider the simultaneous Pell equations $x^{2} - a y^{2} = 1, z^{2} - b x^{2} = 1,$ where $a > b \geq 2$ are positive integers. We describe a procedure which, for any fixed $b$ ,…

math.NT2024

On a conjecture of Levesque and Waldschmidt II

Tobias Hilgart, Volker Ziegler

Related to Shank's notion of simplest cubic fields, the family of parametrised Diophantine equations, \[ x^3 - (n-1) x^2 y - (n+2) xy^2 - 1 = \left( x - λ_0 y\right) \left(x-λ_1 y\…

math.NT2023

Twisted Thue equations with multiple exponents in fixed number fields

Tobias Hilgart, Volker Ziegler

Let $K$ be a number field of degree $d \geq 3$ and fix $s$ multiplicatively independent algebraic integers $γ_{1}, \dots, γ_{s} \in K^{*}$ that fulfil some technical requirements, which c…

On the unique solvability of the simultaneous Pell equations x2−ay2=1 and z2−bx2=1

On a conjecture of Levesque and Waldschmidt II

Twisted Thue equations with multiple exponents in fixed number fields

On the unique solvability of the simultaneous Pell equations $x^{2} - a y^{2} = 1$ and $z^{2} - b x^{2} = 1$