5 problems

Let $F$ be a decomposable form of degree $d$ in $n$ variables of finite type, and let $N_F(m)$ denote the number of integer solutions to the corresponding inequality, while $V(F)$…

Five-prime Linnik representation conjecture. For every fixed $1<c<c_0$ and every sufficiently large positive number $N$, the displayed Diophantine inequality has a solution in prim…

Let $K$ be a number field and let $I\subset {\mathcal O}_K$ be an integral ideal. Define its radical by … where the product is over prime ideals dividing $I$ and…

Let $H(c)$ denote the least number of prime variables needed so that the relevant Diophantine inequality involving powers $p^c$ is solvable, and let $c$ be a real number. Three-pri…

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…