关于奇完全数的研究

关于奇完全数的研究

姓名：XXX 专业班级：信息与计算科学2005XXXXXX 指导教师：XXX

摘要

本文首先介绍了完全数的一些基本性质和当前研究状况，鉴于偶完全数与梅森素数一一对应的特殊关系,接着对梅森素数进行了介绍。

完全数各因子（除1）的倒数和等于1，也就是有若干个循环小数相加，它们的和是1。于是本文又对循环小数的性质进行了讨论，并得出了可喜的结果：两个循环节位数不相等的小数相加，它们的和不会等于1；偶完全数的非2幂因子项的倒数的循环节位数相等。在这个过程中意外的得到了“一个素数，只要非2与5，那么它就会整除一个全1数”。

迄今为止,人类共发现46个完全数,且均为偶完全数.是否有奇完全数存在,至今尚未解决。本文在奇完全数存在的条件下,研究了奇完全数的各因子倒数循环节的规律，得到两个性质：奇完全数的各因子倒数的循环节位数不会是互异的素数；奇完全数的各因子倒数的循环节位数不会相等。

【关键词】完全数；梅森素数；循环节；奇完全数

Study on the Odd Perfect Number

Abstract：This thesis firstly introduce some of the basic nature and current research status of the Perfect Number.In view of Even Perfect Number correspondence with Mersenne prime,then Mersenne prime to have been introduced.

The toal multiplicative inverse of all factor (except 1)of Odd Perfect Number equal 1,in other words,some recurring decimal for adder,the sum equal 1.Then reserth on the recurring decimal,have some encouraging conclusions:if two recurring decimal for adder,have unequal recurrent length,then sum of them can't equal 1; Even Perfect Number non-2 factor have equal recurent length.And have a surprise conclusion:a prime,if it is not 2、5,can divide a all 1 number.

So far, 46 perfect numbers have been found, and they are all Even Perfect Numbers. It is not known whether or not there exists an Odd Perfect Number. In the paper, on the supposition that Odd Perfect Number do exist,give two conclusions:the length of factor's multiplicative inverse of Odd Perfect Number can't all prime number,and can't all equal!

Keywords: perfect number; Mersenne prime; recurrent number; odd perfect number

目录

符号说明.................................................................................................................... - 1 - 第1章前言.............................................................................................................. - 2 - 第2章预备知识...................................................................................................... - 5 - 第3章梅森素数...................................................................................................... - 7 -

3.1有关概念、定理.......................................................................................... - 8 -

3.2 梅森素数判定法的算法设计..................................................................... - 8 -

3.3有关梅森素数分布规律的研究.................................................................. - 9 -

3.4现今的46个梅森素数...............................................................................- 10 - 第4章循环小数.....................................................................................................- 12 - 第5章奇完全数.....................................................................................................- 19 - 结论.........................................................................................................................- 21 - 致谢.........................................................................................................................- 22 - 参考文献...................................................................................................................- 23 -