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
1 6 + 1 3 + 1 2 = 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 = 1 28 + 1 14 + 1 7 + 1 4 + 1 2 {\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]
p n = 2 p − 1 ( 2 p − 1 ) = 2 p − 1 M p , p ∈ P , ( 2 p − 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} }
2 p − 1 ( 2 p − 1 ) {\displaystyle 2^{p-1}(2^{p}-1)} 之因數有
{ 1 , 2 , 4 , e t c , 2 p − 1 , M p , 2 M p , 4 M p , e t c , 2 p − 1 M p {\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
S p + T p = M p + ( 2 p − 1 ) M p {\displaystyle S_{p}+T_{p}=\mathrm {M} _{p}+(2^{p}-1)\mathrm {M} _{p}}
= 2 p M p {\displaystyle =2^{p}\mathrm {M} _{p}}
= 2 ⋅ 2 p − 1 M p {\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?