To the moduli with special forms (such as Mersenne numbers, pseudo-Mersenne numbers, generalized Mersenne numbers), we analysis and study modular arithmetic laws, and obtain the conclusions as follows: for Mersenne numbers and pseudo-Mersenne numbers, we get modular arithmetic formulas, and determine exact expressions of the number of modular addition to generalized Mersenne numbers generated by irreducible monic trinomials and pentanomials.

针对具有特殊形式的模数(如Mersenne数，伪Mersenne数，广义Mersenne数等)，深入分析了其取模运算规律，得到如下结果：对Mersenne数和伪Mersenne数，给出了取模运算转换为模加或模减运算的公式；对模数为任意首一三项式和五项式产生的广义Mersenne数，推导了相应取模运算的复杂度计算解析表达式。

By replacing pseudo-Mersenne numbers with generalized Mersenne numbers, we propose a new notion-Generalized Optimal Extension Fields, and study the fast arithmetic about multiplication and modular arithmetic in GOEFs.

利用广义Mersenne数代替伪Mersenne数，提出了广义最优扩域的概念，并研究其上的快速乘法运算和取模运算，为乘法运算给出了通用的复杂度公式，为取模运算给出了具体的运算公式，推广了Bailey，Mihailescu和Woodbury等在最优扩域上的相应结果。

Some of the oldest open problems in mathematics relate to the abundancy index of a number. Are there infinitely many perfect numbers? Does there exist an odd perfect number? Does there exist an odd multiply perfect number? Such questions have been considered for well over two millennia by the likes of Euclid, Fermat, Descartes, Mersenne, Legendre, and Euler.

暑期的数学研究已渐渐步入正轨，今天上午和导师讨论的时候提出了第一个conjecture，说是虽然6和12的倍数的abundancy index比较高，但是如果把两个相邻的数字相乘，他们乘积的abundancy index总体来说会甚至更高。

meromorphic function：半纯函数

Merlin 操作系统 Merlin | 半纯函数 meromorphic function | Mersenne 质数 Mersenne prime

实数,有理数? irrational(number) 无理数：real number, rational number

negative whole number 负整数?? consecutive number 连续整数 | real number, rational number 实数,有理数? irrational(number) 无理数 | absolute value 绝对值

negative whole number：负整数?? consecutive number 连续整数

positive whole number 正整数 | negative whole number 负整数?? consecutive number 连续整数 | real number, rational number 实数,有理数? irrational(number) 无理数