YLL討論網

由 **◤呆‧呆◢** 於星期二八月 15, 2006 9:47 am

#ed_op#DIV#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#十三#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證明中的錯誤逐一被發現#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#更多#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證明中的錯誤逐一被發現。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1911#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Power#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#找出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#88#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#89#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是完美數，幾年後他又找出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#101#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，因此#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#100#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#101#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是一個完美數。於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1922#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Kraitchik#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#發現了，關於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數最大為#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#257#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#之證明是錯誤的，因他證出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#257#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#不是質數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#十四#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#結論#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#我們已逐一找出偶完美數，但我們更希望證明奇完美數不可能存在。目前研究的主要方法，是找出奇完美數的最少相異質因數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#,#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#而且奇完美數是存在的。於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1888#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Sylvester#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#發現任一奇完美數，至少有#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#4#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#個相異質因數。不久，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Sylvester#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#自己修正這項結論，認為任一奇完美數，至少有#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#5#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#個相異質因數。直到今天我們已知，至少需有#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#個相異質因數，或至少有#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#29#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#個不一定相異的質因數，方能構成奇完美數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#SPAN#ed_cl#至今已找出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#37#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#個完美數，其中#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#88#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#89#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是最後一個經由人工計算而獲得的完美數，其餘都是利用電子計算機找出的。事實上電子計算機，對#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數和完美數的發現，帶來一項新趣味。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1998#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年九月，筆者寫這篇文章時，已知最大#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#3021377#ed_op#/SUP#ed_cl# -1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，也就是說最大完美數是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#3021376#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#3021377#ed_op#/SUP#ed_cl# –1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。此最大完美數共有#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1819050#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#位數。#ed_op#/SPAN#ed_cl##ed_op#/DIV#ed_cl#

由 **◤呆‧呆◢** 於星期二八月 15, 2006 9:45 am

#ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#一#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#前言#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#究竟人類從何時開始研究完美數，我們並不知道。但埃及人，已經很自然的用它來計算了。而畢達哥拉斯〈#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Pythagoras#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#〉和他的門徒研究完美數的神秘色彩卻多於數的理論性質。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#比較早對完美數的定義是關於整除的部份，當某數等於其所有因數和時，它即是完美數。例如#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl# 1=10/10#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2=10/5#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#5=10/2#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#但是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl# 10≠1+2+5#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#所以#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl# 10#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#不是完美數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#而最早找出的四個完美數是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl# 6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#28#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#496#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#和#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8128#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，發現者已無法考證。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl# #ed_op#SPAN#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6=1+2+3#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#28=1+2+4+7+14#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#496=1+2+4+8+16+31+62+124+248#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8128=1+2+4+8+16+32+64+127+254+508+1016+2032+4064#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#BR#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#二#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#歐幾里得幾何原本關於完美數的記錄#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#西元前#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#300#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，在歐幾里得幾何原本是數學史上最早有關於完美數的記錄。因它是記錄在幾何原本中，所以讓人感到相當驚訝！其記錄在幾何原本第九冊性質#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#36#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，敘述如下：#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#從#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#開始一直#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#倍的相加，直到總和為質數時，此質數再乘以最後相加的數，即為完美數。例如：#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1+2+4=7#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，因為#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#7#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，所以〈總和〉#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#*#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#〈最後一個數〉#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#=7*4=28#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#28#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#就是完美數。又如#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1+2+4+8+16=31#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，因為#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#31#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，所以#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#31*16=496#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#496#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#就是完美數。因此綜合得公式如下：#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1+2+4+8+……+2#ed_op#SUP#ed_cl#k-1#ed_op#/SUP#ed_cl# =2#ed_op#SUP#ed_cl#k#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#若#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#k>1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#k-1#ed_op#/SUP#ed_cl##ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數時，則#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#k-1#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#k#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#就是完美數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#三#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#希臘的#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Nicomachus#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#對完美數的討論#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#於西元#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#100#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，希臘的#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Nicomachus#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是第二位嚴謹地討論完美數者，在他的名著#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Introductio Arithmetica#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#中，將數分成三類：#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#第一類：過剩的數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#─#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#整除部分的和超過自己。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#第二類：不足的數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#─#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#整除部分的和小於自己。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#第三類：完美的數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#─#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#整除部分的和等於自己。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#後來#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Nicomachus#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#雖未經證明，仍推論出關於完美數的部分性質。以現代的定義敘述如下：#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(1)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#第#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#n#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#個完美數有#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#n#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#位數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(2)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#所有完美數皆為偶數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(3)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#所有完美數的尾數都是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#或#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#這二數交替。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(4)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#以歐幾里得生成完美數之公式，即可得所有完美數。換句話說，當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl# k > 1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，而#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#k#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#是質數時，每一個完美數的型態都是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#k-1#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#k#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(5)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#完美數有無限多個。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#接下來我們將說明後來有哪些人證明#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#和#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(3)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是錯誤的，而#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(2)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(4)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#和#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(5)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#仍是問題。雖然#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(1)~(5)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#項很多人當成真理，事實上並未被證實，如完美數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#28#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#496#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8128#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#使人們誤以為#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(3)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是正確的。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Saint Augustine(354-430)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#在其名著#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#The City of God#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#中更寫道#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#─#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#６這個數本身即是完美數，並不是因為在６天內上帝就創造了萬物；相反的是因為６這個數是完美的，所以上帝才在#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#天內創造萬物。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#四#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#阿拉伯數學家對完美數的研究#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#阿拉伯數學家對完美數亦相當著迷，其中#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Thabit ibn Qurra#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#驗證出，當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數時，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#n#ed_op#/SUP#ed_cl# p#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的型式必是完美數。而#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Ibn al-Haytham#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#不但提出歐幾里得性質的逆命題，並證明了當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#k#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數時，型如#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#k-1#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#k#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#必是完美數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#五#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#歐洲數學家對完美數的研究#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1500#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年來歐洲數學家皆認為#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Nicomachus#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的推論是正確的，甚至有數學家相信另一未被證實而且錯誤的結論#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#─#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#k#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是奇數時，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#k-1#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#k#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是完美數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1461#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年第五個完美數被發現，過不久第六個完美數也被發現了，但未能確定發現者是誰。直到#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1536#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Hudalrichus Regiusg#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是第一位推翻對於後來數學家認為是常識的#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Nicomachus#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#性質，在他的著作#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Utriusque Atithmetics#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#中指出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#11#ed_op#/SUP#ed_cl#-1=2047=23*89#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。由此發現第一個質數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，使得#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p-1#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#不是完美數。他又證明了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#13#ed_op#/SUP#ed_cl#-1=8191#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，所以發現第五個完美數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#12#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#13#ed_op#/SUP#ed_cl#-1)=33550336#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。而第五個完美數卻有#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#位數，證明了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Nicomachus#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的第一個推論是錯誤的。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#六#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl# 1603#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Cataldi#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#對完美數的研究#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#到了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1603#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Cataldi#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#利用它的質數表，發現#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#17#ed_op#/SUP#ed_cl#-1=131071#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，所以發現第六個完美數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#16#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#17#ed_op#/SUP#ed_cl#-1)=8589869056#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。而第五個及第六個完美數的尾數是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，這項結果證明了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Nicomachus#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的第三個推論是錯誤的。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Cataldi#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#又利用它的質數表，發現#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl# 2#ed_op#SUP#ed_cl#19#ed_op#/SUP#ed_cl#-1=524287#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，所以發現第七個完美數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#18#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#19#ed_op#/SUP#ed_cl#-1)=137438691328#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。而第五個及第六個完美數的尾數是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，這項結果證明了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Nicomachus#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的第三個推論是錯誤的。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Cataldi#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#雖然具有發現二個完美數的重大成就，但他仍下錯結論。在其著作#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Utriusque Arithmetices#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#中寫著指數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p=2#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#3#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#5#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#7#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#13#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#17#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#19#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#23#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#29#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#31#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#37#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#時，使得#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p-1#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是完美數。當然，因為他已經利用他的質數表證明了當指數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p=2#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#3#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#5#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#7#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#13#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#17#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#19#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#時，是正確的；但是他另外四個證明#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#23#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#29#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#31#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#37#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#中，卻只有一個是對的。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#七#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl# 1638#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Descartes#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#對完美數的研究#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#許多數學家對完美數相當有興趣，並嘗試著提出理論。如在#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1638#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Descartes#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#曾寫信給#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，寫著：#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#我想我可以由歐幾里得的公式證明，沒有不是偶數的完美數；但奇數的完美數必是某一質數乘以一完全平方數，且其平方根為質數的合成。如#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#22021#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#9018009#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的平方根是由質數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#3#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#7#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#11#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#13#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的合成，所以#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#22021*9018009=198585576189#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是完美數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#…#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#八#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#對完美數有重大貢獻的#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Fermat#ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#下一個對完美數有重大貢獻的是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Fermat#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1640#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#月他寫信給#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#告訴他關於完美數他的發現，寫著：#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#…#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#我已經發現以下三項性質，首先定義完美數的根數，由指數所形成如下，其中#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#3#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#4#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#5#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#等是指數，下一行則為根數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#TABLE style="BORDER-RIGHT: medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-COLLAPSE: collapse" cellSpacing=0 cellPadding=0 border=1#ed_cl##ed_op#TBODY#ed_cl##ed_op#TR#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT: windowtext 1pt solid; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#3#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#4#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#5#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#7#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#9#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-TOP: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New 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style="BORDER-RIGHT: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BORDER-TOP-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#255#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BORDER-TOP-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#511#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BORDER-TOP-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1023#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BORDER-TOP-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2047#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BORDER-TOP-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#4095#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#TD style="BORDER-RIGHT: windowtext 1pt solid; BORDER-LEFT-COLOR: #ece9d8; BORDER-TOP-COLOR: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8191#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#/TD#ed_cl##ed_op#/TR#ed_cl##ed_op#/TBODY#ed_cl##ed_op#/TABLE#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(1)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#當指數是合成數時，他的根數即是合成數。如#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#63#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的指數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是合成數，所以我們就說#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#63#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是合成數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(2)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#當指數是質數時，他的根數減#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#可以被指數的兩倍整除。如#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#127#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的指數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#7#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#126#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#可以是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#14#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的倍數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(3)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#當指數是質數時，他的根數不能被任何其他的質數整除。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#以上三項漂亮性質，我將他稱為完美數的基本定理。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New 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style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(1)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#如果#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#n#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是合成數時，則#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#n#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#即是合成數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(2)#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#如果#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#n#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數時，則#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#n#ed_op#/SUP#ed_cl#-2#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New 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Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#九#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl# Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#對完美數的結論#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#對#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Fermat#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#寄給他關於完美數的結論相當感興趣，而且很快提出他自己的證明，使許多數學家著迷好多年。於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1644#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，他出版#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Cogitata physica mathematica#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#書中證明了當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p=2#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#3#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#5#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#7#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#13#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#17#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#19#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#31#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#67#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#127#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#257#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#時，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數且#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p-1#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#也是完美數。所以型如#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的質數，稱為#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#十#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧#ed_op#SPAN#ed_cl# #ed_op#/SPAN#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#對完美數有重大貢獻的#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Euler#ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#下一個對完美數有重大貢獻的是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Euler#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1732#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年他發現第八個完美數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#30#ed_op#/SUP#ed_cl# (2#ed_op#SUP#ed_cl#31#ed_op#/SUP#ed_cl#-1)=2305843008139952128#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。這是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#125#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年來第一個被發現的完美數。接著於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1738#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，尤拉解決了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Cataldi#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的結論最後一項錯誤，因為#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#29#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#不是質數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#在他兩份於生前未發表的手稿中，尤拉證明了歐幾里得的公式之逆命題，每一個偶完美數的型式必是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p-1#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#；由此也很容易的導出所有偶完美數的尾數是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#6#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#或#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#8(#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#但並不是交替出現#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。尤拉更嘗試著找出奇完美數是否存在？後來尤拉利用在#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1638#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Descartes#ed_op#/SPAN#ed_cl# #ed_op#SPAN#ed_cl#寫給#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#信中所提的結論，證明了每一個奇完美數，當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#4n+1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數時，其型式必是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#(4n+1)#ed_op#SUP#ed_cl#4k+1#ed_op#/SUP#ed_cl#b#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#²#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。他也提出了當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p=41#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#和#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p=47#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#時，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p-1#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是完美數，但是在#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1753#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，他發現了自己錯誤的性質並修正之。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#對於完美數的研究，已經變成企圖去檢驗，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#在他#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Cogitata physica mathematica#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#書中證明的是否正確了。而尤拉更留下了將近#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#150#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年來，最大的完美數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#30#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#31#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。其他數學家像#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Peter Barlow#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1811#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年在他#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Theory of Numbers#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#書中寫著：#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#…#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#完美數#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#30#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#31#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，是已發現的完美數中最大的。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#十一#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧1876#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Lucas#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#發現了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證明中第一個錯誤#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1876#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Lucas#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#發現了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證明中，第一個錯誤。雖然他的方法無法找出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#67#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的任何因數，但他卻能證明#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#67#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#不是質數。同時，當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Lucas#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證明#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#127#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#的質數，和#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#126#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#127#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是一個完美數時，也就是說#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Lucas#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#發現了#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數中，有一個是正確的。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Lucas#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#另有一項重大貢獻，利用電子計算機找尋#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數，也就是找完美數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#十二#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧Catalan#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#數列#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Lucas#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證明#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2127 –1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數之後，緊接著#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Catalan#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#猜測，如果#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#m=2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，則#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#m#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#也是質數。當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p=3#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#7#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#127#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#170141183460469231731687303715884105727#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#時，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Catalan#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#數列就是#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#。當然如果這項猜測是正確的，就可以解決是否有無限個#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數，這個至今仍未解決的問題。但不管用什麼方法驗證，當#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#p=170141183460469231731687303715884105727#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#時，第四組#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Catalan#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#數列#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#p#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是不是質數，已超出我們能力範圍。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1883#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Pervusin#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#60#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#61#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是完美數。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1903#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Col#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#發現了，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Lucas#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#曾證明#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#67#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是合成數，並找出該數的因數。於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1903#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#10#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#月，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Col#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#參加#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#the American Mathematical Society#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#會議時，發表一篇論文提出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#67#ed_op#/SUP#ed_cl#-1=147573952589676412927=761838257287*193707721#ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#獲得觀眾熱烈掌聲。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P align=center#ed_cl##ed_op#FONT size=3#ed_cl##ed_op#SPAN#ed_cl#十三#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#‧Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證明中的錯誤逐一被發現#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl##ed_op#/FONT#ed_cl##ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl##ed_op#/SPAN#ed_cl# #ed_op#/P#ed_cl##ed_op#P#ed_cl##ed_op#SPAN#ed_cl#更多#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#證明中的錯誤逐一被發現。#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1911#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Power#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#找出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#88#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#89#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是完美數，幾年後他又找出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#101#ed_op#/SUP#ed_cl#-1#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是質數，因此#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#100#ed_op#/SUP#ed_cl#(2#ed_op#SUP#ed_cl#101#ed_op#/SUP#ed_cl#-1)#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#是一個完美數。於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#1922#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#年，#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Kraitchik#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#發現了，關於#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#Mersenne#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#質數最大為#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#257#ed_op#/SPAN#ed_cl##ed_op#SPAN#ed_cl#之證明是錯誤的，因他證出#ed_op#/SPAN#ed_cl##ed_op#SPAN style="FONT-FAMILY: 'Times New Roman'"#ed_cl#2#ed_op#SUP#ed_cl#257#ed_op#/

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