13 problems

Let $P_j\in\mathbb{Z}[u]$ for $j=1,2,\ldots,d-1$, and consider the parametric Thue equation … The trivial solutions are $(1,0)$, $(0,1)$, and $(P_j,1)$ for $j\in\{1,2,\ldots,d-1\}$…

Let $p_2(t),\ldots,p_d(t)\in\mathbb{Z}[t]$ be monic polynomials, and consider the split family of Thue equations … Its trivial solutions are … where $\epsilon=1$ for odd $d$ and…

Levesque–Waldschmidt conjecture. There exists a positive absolute constant $\kappa$ such that, whenever

Let $F\in\mathbb Z[x,y]$ be a binary form with nonzero discriminant and degree at least three. For an integer $m$, consider the Thue equation … A solution $(x,y)$ is primitive when…

Let $F\in\mathbb Z[x,y]$ be a homogeneous binary form of degree $d\geq 3$ with nonzero discriminant, let $m$ be a fixed positive integer, and consider the Thue equation … A solutio…

Let $f(x,y)$ be an integral binary quartic form of nonzero discriminant. Integer solutions $(x,y)$ to the quartic Thue equations … are considered up to the identification…

The paper considers the relative Thue equations and the corresponding parametrizations of solutions described in Theorems 3 and 4, with parameters $t$ and variables $x,y,z$ as spec…

Folk conjecture. A density of $100\%$ of integral binary $n$-ic forms that locally represent $h$ do not globally represent $h$, meaning that $F(x,y)=h$ has no solution in…

Rank-two unit conjecture. There exists a positive absolute constant $\kappa$ such that, whenever $n,s,t,x,y,m$ are integers satisfying

Soit $\alpha$ un nombre algébrique de degré $d\geq 3$ sur $\mathbb{Q}$, soit $K=\mathbb{Q}(\alpha)$, et soit $\mathbb{Z}_K^\times$ le groupe des unités de $K$. Pour toute unité…

Siegel's conjecture. An upper bound for the number of primitive solutions should be obtainable that is independent of the particular coefficients of the polynomial $F$. The conject…

Let $B,n\in\mathbb{N}_{>1}$ satisfy … Consider the equation … where solutions with $Z\in\{-1,0,1\}$ are called trivial. Unique-solution conjecture. Under these conditions, the equa…

Let $\alpha$ be an algebraic number of degree $d\geq 3$ over $\mathbf{Q}$, let $K=\mathbf{Q}(\alpha)$, and let $\mathbf{Z}_K^\times$ be the group of units of $K$. For a unit…