八的三角定理与二次多项式表示

高斯三角定理的一个推广

A generalization of Gauss' triangular theorem

一个二次多项式$Φ_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$称为泛多项式，如果丢番图方程$Φ_{a,b,c}(x,y,z)=n$对任意非负整数$n$都有整数解$x,y,z$.在这篇文章中，我们表明，如果$(a,b,c)=(2,2,6), (2,3,5)$或$(2,3,7)$，那么$Φ_{a,b,c}( x,y,z)$是通用的。这些是Sun在\cite {Sun}中记录的。

The number of representations by a ternary sum of triangular numbers

For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some relations between $t(a,b,c;n)$ and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in \cite{s}.

三角数的偶泛和

Even universal sums of triangular numbers

本文分类了所有偶泛的三角数之和，并给出了任意三角数之和偶泛性的有效判定准则，推广了 Bosma 和 Kane 的三角数八定理。

实二次函数域的理想类群与子群

Ideal class groups and subgroups of real quadratic function fields

本文研究了代数函数域K，给出了实二次函数域$K$的理想类群$H(K)$具有$n$阶循环子群的充要条件，得到了八个这样的域系列$K$，其中四个不是ERD型或GERD型.

Partitions Associated to Class Groups of Imaginary Quadratic Number Fields

We investigate properties of attainable partitions of integers, where a partition $(n_1,n_2, \dots, n_r)$ of $n$ is attainable if $\sum (3-2i)n_i\geq 0$. Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian $p$-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes $p$. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.

When are Multiples of Polygonal Numbers again Polygonal Numbers?

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as …

正三元二次多项式表示整数

The representation of integers by positive ternary quadratic polynomials

一个整数二次多项式被称为正则多项式，如果它表示由多项式本身在实数和$p$ -进整数上对每个素数$p$表示的每个整数。如果它的形式为$Q({\mathbf x} + {\mathbf v})$，则称为完备的，其中$Q$是变量${\mathbf x} = (x_1, \ldots, x_n)$的整二次型，${\mathbf v}$是${\mathbb Q}^n$中的向量。它的导体被定义为最小正整数$c$，使得$c{\mathbf v} \in {\mathbb Z}^n$。证明了对于一个固定的正整数$c$，正本原三元正则完全二次多项式与导体$c$的等价类只有1000个.这推广了Watson关于正定正则三元二次型和Chan和Oh关于三元三角形的类似有限性结果。

八次代数数的四个共轭的线性关系

Linear relations of four conjugates of an algebraic number of degree eight

刻画了存在四个不同共轭满足线性关系α1=α2+α3+α4的八次代数数，并得到类似刻画α1+α2+α3+α4=0。特别地，当八次代数数的极小多项式非偶且恰有六个形如α_i1+α_i2+α_i3+α_i4=0的不同线性关系时，证明该数为二次与四次代数数之和。

有限域上全三角多项式的根计数

Counting roots of fully triangular polynomials over finite fields

本文研究有限域上全三角多项式的根计数问题，并在其增广次数矩阵与线性或对角二次多项式行等价时给出根数量的显式公式。

Vaaler's theorem in number fields

In Diophantine approximation, Vaaler's theorem was an important partial result towards the Duffin--Schaeffer conjecture, which was open for almost eighty years before it was recently proven by Koukoulopoulos and Maynard. A version of this result was previously proven to also hold in imaginary quadratic fields: in this paper, we establish a version of Vaaler's theorem in general number fields.

虚二次域的类数ζ函数

Class number zeta function of imaginary quadratic fields

本文引入一个通过类数计数虚二次数域的ζ函数，证明该函数仅依赖于八次单位根，从而得到素数类数的下界。方法基于动力系统周期点的研究。

关于几乎泛混合平方和与三角数

On almost universal mixed sums of squares and triangular numbers

1997年K. Ono和K. Soundararajan [发明。Math.130（1997）]证明了在广义黎曼假设下，任何大于2719的正奇数都可以用著名的拉马努金形式$x^2+y^2+10z^2$表示，等价地，形式$2x^2+5y^2+4T_z$表示所有大于1359的整数，其中$T_z$表示三角数$z(z+1)/2$。给定正整数$a,b,c$，我们采用模形式和二次型的理论，以确定完全时，一般形式$ax^2+by^2+cT_z$表示足够大的整数，并建立类似的结果，形式$ax^2+bT_y+cT_z$和$aT_x+bT_y+cT_z$。下面是我们的主要定理的一些结果：（1）所有足够大的奇数都有$2ax^2+y^2+z^2$的形式当且仅当$a$的所有素因子全等于1模4。(ii)形式$ax^2+y^2+T_z$几乎是通用的（即，它表示足够大的整数）当且仅当$a$的每个奇素因子全等于1或3模8。(iii)$ax^2+T_y+T_z$几乎是泛的当且仅当$a$的所有奇素因子全等于1模4。(iv)当$v_2(a)\not=3$时，形式$aT_x+T_y+T_z$几乎是普适的当且仅当$a$的所有奇素因子全等于1模4和$v_2(a)\not=5,7,...$，其中$v_2(a)$是$a$的2进阶。