[轉貼]完美數的故事(續)
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#