报告地点：行健楼学术活动室526
报告人简介：
洪绍方，四川大学数学学院教授、博士生导师，教育部新世纪优秀人才，四川省学术和技术带头人。先后负责主持国家自然科学基金和教育部博士点基金等10多个纵向项目。已经在国内外数学期刊发表学术论文100多篇，培养毕业硕士60多名，毕业博士20多名，其中多人已晋升正高职称。
报告摘要：
Let $n\ge 1, e\ge 1, k\ge 2$ and $c$ be integers. An integer $u$ is called a unit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(u, n)=1$. A unit $u$ is called an exceptional unit in the ring $\mathbb{Z}_n$ if $\gcd(1-u,n)=1$. We denote by $\mathcal{N}_{k,c,e}(n)$ the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^e+...+x_k^e\equiv c \pmod n$ with all $x_i$ being exceptional units in the ring $\mathbb{Z}_n$. In 2017, Mollahajiaghaei presented a formula for the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^2+...+x_k^2\equiv c\pmod n$ with all $x_i$ being the units in the ring $\mathbb{Z}_n$. Meanwhile, Yang and Zhao gave an exact formula for $\mathcal{N}_{k,c,1}(n)$. In this talk, by using Hensel's lemma and the techniques of exponential sums as well as quadratic Gauss sums, we derive an explicit formula for the number $\mathcal{N}_{k,c,2}(n)$. Our result extends Mollahajiaghaei's theorem and that of Yang and Zhao.
