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ܼ

[]nФơnȤ٤Ƥ¤nȰפȤnȤ

[]n٤Ƥ¤(n)Ƚ񤯡

[]nǤȤ(n)=2nΩġ

[]n=6ϴǤ롣ʤʤС61,2,3,6Ǥꡢ6Ȥ­ȡ1+2+3=6Ȥʤ뤫Ǥ롣

  • ʺաˡ(6)=1+2+3+6=(1+2+3)+6=6+6=12
  • ʱաˡ26=12

  • γǰϡظ٤7Ƥ롣
    • ˴򰷤äʸΤƤʤ
  • 100ǯ˥˥ޥ󥳥ˤäƽ񤫤줿ܤˤ⡢Ȥ6,28,496,8128󤲤Ƥ롣
  • p=11б2096127(=(211-1)211-1)ϤФиäƴȤ줿2047(=211-1)ǿǤϤʤ
    • 2047=2389

ȥ顼ؿ

ǰʬ򤬤狼äƤ뼫Ф(n)η׻

[]nǰʬn=paqbrcĤȤС(n)ϼμǷ׻Ǥ롣
\sigma(n)=\frac{p^{a+1}~-~1}{p-1}~\cdot~\frac{q^{b+1}~-~1}{q-1}~\cdot\frac{r^{c+1}~-~1}{r-1}~\cdots

[]˼Ѥͤ롣

(1+p++pa)(1+q++qb)(1+r++rc)

Ÿݤ˸ƹpqrĤˤꡢnʽʣʤˤ٤Ϳ롣
ʤʤС[]ֹnǰʬn=paqbrcĤȤСΤpqrġ0a0¡b0ácĤͿפΩĤ顣

äơѤͤϦ(n)

(n)=(1+p++pa)(1+q++qb)(1+r++rc)

ˡθ

1+p+\cdots+p^a=\frac{p^{a+1}~-~1}{p-1}

ŬѤСΩġ

\sigma(n)=\frac{p^{a+1}~-~1}{p-1}~\cdot~\frac{q^{b+1}~-~1}{q-1}~\cdot\frac{r^{c+1}~-~1}{r-1}~\cdots

[]n=496ɤĴ٤롣

nǤС֦(n)=2nפΩĤᡢޤ(n)׻롣

496=2431ǰʬǤΤǡ(496)ϼΤ褦ˤƷ׻Ǥ롣

(496)
=\frac{2^{4+1}-1}{2-1}~\cdot~\frac{31^{1+1}-1}{31-1}
=(32-1)(31+1) =3125 =2496

äơ496ϴǤ롣

륻̿

[]
pǿMp륻ǿMp=2p-1ˤȤ롣
ΤȤMp2p-1͡ʴˤʤˤ1ΰ̤6 or 8Ǥ롣

[]2Τ٤ϼΤ褦ˤʤ롣

2,22=4,23=8,24=16,25=32

äơ2p-11ΰ̤2,4,6,8Τɤ줫Ǥ롣

2p-12468
2p4826嵭ιԤ2ܤ̤1ΰ
2p-11ΰ3715륻ǿθ䡣嵭ιԤ-1򤷤̤1ΰ
(2p-1)2p-16860־ιԤ1ľιԤ軻̤1ΰ

2p-11ΰ̤ο5ξˤϡ2p-1ǿǤϤʤ
ʤʤС1ΰ̤ο5ǿǤϡ5ΤȤꤨʤ2p-1=5p¸ߤʤǤ롣
äơ2p-1ϥ륻ǿǤϤʤΤǡ롣

äơMp2p-1ͤ1ΰ̤6 or 8Ǥ롣

ζ

  • δϴ˷ޤäδ̤1Ĥ⸫ĤäƤʤ
  • δϡпξͤФʤʤȤΤƤơ餯¸ߤʤȹͤƤ뤬̤Ǥ롣

[]̤
δ¸ߤʤ

[]¸ߤС10300ʾοǤ롣

̵

[]̤
̵¸¸ߤ롣

[]륻ǿ̵¤ˤ뤫狼äƤʤ⤷̵¤ˤСδ̵¤ˤ뤳Ȥˤʤ롣

ʸ

  • عʷϤΤν硡Źᤶơ
  • ؤʤäȤ륪顼ȥեޡ