完美數者,正整數之同於其因數和也,且不含其自身。亦可云遺傳子數和二倍於基自身。
以六爲例,有
6=1+2+3{\displaystyle 6=1+2+3} ①
2×6=1+2+3+6{\displaystyle 2\times 6=1+2+3+6} ②
令①除六,得
16+13+12=1{\displaystyle {\frac {1}{6}}+{\frac {1}{3}}+{\frac {1}{2}}=1}
廿八亦此
28=1+2+4+7+14{\displaystyle 28=1+2+4+7+14}
同除廿八得
1=128+114+17+14+12{\displaystyle 1={\frac {1}{28}}+{\frac {1}{14}}+{\frac {1}{7}}+{\frac {1}{4}}+{\frac {1}{2}}}
6, 28, 496, 8128, 33550336, etc
亦可見list of perfect numbers, emwiki[一]
pn=2p−1(2p−1)=2p−1Mp ,p∈P ,(2p−1)∈P{\displaystyle {\mathfrak {p}}_{n}=2^{p-1}(2^{p}-1)=2^{p-1}\mathrm {M} _{p}\ ,p\in \mathbb {P} \ ,(2^{p}-1)\in \mathbb {P} }
2p−1(2p−1){\displaystyle 2^{p-1}(2^{p}-1)} 之因數有
{1,2,4,etc,2p−1,Mp,2Mp,4Mp,etc,2p−1Mp{\displaystyle 1,2,4,\mathrm {etc} ,2^{p-1},\mathrm {M} _{p},2\mathrm {M} _{p},4\mathrm {M} _{p},\mathrm {etc} ,2^{p-1}\mathrm {M} _{p}} }
則
Sp+Tp=Mp+(2p−1)Mp{\displaystyle S_{p}+T_{p}=\mathrm {M} _{p}+(2^{p}-1)\mathrm {M} _{p}}
=2pMp{\displaystyle =2^{p}\mathrm {M} _{p}}
=2⋅2p−1Mp{\displaystyle =2\cdot 2^{p-1}\mathrm {M} _{p}}
即其因數和二倍于其自身,故得證之。
(2p-1)者,梅森質數也,P{\displaystyle \mathbb {P} } 者,質數集也,若無歧義,亦可書P{\displaystyle \mathrm {P} } .
由完美數公式可知,尋梅森質數即尋完美數,計算機未發明之時,則其甚難尋之,今有GIMPS之項[二],故其之尋有所破,然仍有二疑:奇完美數之存乎?完美數無窮乎?