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完全數

文出維基大典

完全數者,正整數也。其正因子之和適等於本數。如六,一、二、三之和也;二十八,一、二、四、七、十四之和也。[]

歐幾里得證之:二冪減一為素數,則此素數與前冪之積為偶完全數。如六、二十八、四百九十六、八千一百二十八是也。[]歐拉復證:偶完全數必如是形。[]

迄今未知奇完全數之存否,亦未明完全數之多寡。[]然學者考之,得若干性質:

其一,偶完全數皆為三角數,亦為六邊數[]

其二,偶完全數之二進式表甚特,皆若干一後接等數之零。[]

其三,完全數之因子之倒數和必為二。[]

其四,完全數之因子數必為偶。[]

奇完全數若存,必大於十之千五百冪。其因子至少百有一。[]

數學家窮索數千載,然此數之奧秘尚未盡明。非淺學所能窺其堂奧,唯精於此道者,方能探其精微。[一〇]

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