Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in $\mathbb{R}^2$

An unexpected meeting between the $P^{3}_{1}$-set and the cubic-triangular numbers

A set of $m$ positive integers $\{x_{1},\ldots,x_{m}\}$ is called a $P^{3}_{1}$-set of size $m$ if the product of any three elements in the set increased by one is a cube integer. A $P^{3}_{1}$-set $S$ is said to be extendible if there exists an integer $y\not\in S$ such that $S\cup\{y\}$ still a $P^{3}_{1}$-set. Now, let consider the Diophantine equation $u(u+1)/2=v^{3}$ whose integer solutions produce what we called cubic-triangular numbers. The purpose of this paper is to prove simultaneously that the $P^{3}_{1}$-set $\{1,2,13\}$ is non-extendible and $n=1$ is the unique cubic-triangular nu…

A generalization of Gauss' triangular theorem

A quadratic polynomial $Φ_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation $Φ_{a,b,c}(x,y,z)=n$ has an integer solution $x,y,z$ for any non negative integer $n$. In this article, we show that if $(a,b,c)=(2,2,6), (2,3,5)$ or $(2,3,7)$, then $Φ_{a,b,c}( x,y,z)$ is universal. These were conjectured by Sun in \cite {Sun}.

Diophantine problems in solvable groups

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural "non-commutativity" conditions. For each group $G$ in one of these classes, we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by finite systems of equations (e-interpretable), and hence that the Diophantine problem in $O$ is polynomial time reducible to the Diophantine problem in $G$. One of the major open conjectures in number theory st…

上三角2×2整数矩阵中的$D(N)$四元组

$D(N)$-quadruples in upper-triangular $2\times2$ integer matrices

该研究在非交换环$M_2(\boldsymbol Z)$中引入$D(N)$-$m$元组的类似物，考虑标准矩阵乘积与若尔当乘积两种形式，聚焦上三角整数矩阵$UT_2(\boldsymbol Z)$，探究其中若尔当$D(N)$四元组的存在性。

The Frobenius number for sequences of triangular numbers associated with number of solutions

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer $p$. It is easy to find the explicit form of this number in the case of two variables. However,…

Real Interference Alignment

In this paper, we show that the total Degrees-Of-Freedoms (DOF) of the $K$-user Gaussian Interference Channel (GIC) can be achieved by incorporating a new alignment technique known as \emph{real interference alignment}. This technique compared to its ancestor \emph{vector interference alignment} performs on a single real line and exploits the properties of real numbers to provide optimal signaling. The real interference alignment relies on a new coding scheme in which several data streams having fractional multiplexing gains are sent by transmitters and interfering streams are aligned at recei…