Numerus perfectus

Numerus perfectus est numerus naturalis ${\displaystyle n}$, cuius summa divisorum, numero ipso excepto, numero ${\displaystyle n}$ adaequat. Vel symbolis mathematicis expressum

Systemata Numerica Mathematicae

Numeri Elementarii
Naturales ${\displaystyle \mathbb {N} }$ {0,1,2,3,...} sive {1,2,3,...} Primi ${\displaystyle \mathbb {P} }$ {2,3,5,7,11,...} Abundantes Amicabiles Compositi Defectivi Perfecti Sociabiles Integri ${\displaystyle \mathbb {Z} }$ {...,-2,-1,0,+1,+2,...} Pares {...,-2,0,+2,...} Impares {...,-3,-1,+1,+3,...} Rationales ${\displaystyle \mathbb {Q} }$ Reales ${\displaystyle \mathbb {R} }$ Irrationales Numerus pi ${\displaystyle \pi =3.14159265358979\ldots }$ Numerus Euleri ${\displaystyle e=2.718281828459045\ldots }$ Complexi ℂ Algebraici Transcendentes Numerus imaginarius ${\displaystyle i={\sqrt {-1}}}$ Quaterni ${\displaystyle \mathbb {H} }$ Octoni ${\displaystyle \mathbb {O} }$ Infinitas ${\displaystyle \infty }$
Variae radices
Radix binaria(2) Radix octalis(8) Radix decimalis(10) Radix sedecimalis(16)

${\displaystyle n=\sum _{d|n,d\not =n}d}$.

Exempla:

• ${\displaystyle 6=1+2+3}$
• ${\displaystyle 28=1+2+4+7+14}$

Demonstrabile est omnes numeros pares perfectos hanc speciem habere et quicumque numerus naturalis hanc speciem habet perfectum esse, si utroque casu ${\displaystyle 2^{k}-1}$ est numerus primus Mersenni :

${\displaystyle n=2^{k-1}\cdot (2^{k}-1).}$

Adhuc autem incertum est, an omnino sint numeri perfecti impares.

Nexus interni

• Numerus abundans
• Numerus defectivus