A perfect number is the sum of its proper divisors , by definition .
Perfect Numbers = Perfect Integers = Perfect Naturals = Perfects
p p-1 p
2 - 1 is prime ===> P = 2 ( 2 - 1 ) is perfect
p
E.g. , p = 3 , 2 - 1 = 7 , P = 1 + 2 + 4 + 7 + 14 = 28 .
p
Every even perfect is of this form , where P = Triangle ( 2 - 1 ) ;
e.g. , 28 = 7 + 6 + 5 + 4 + 3 + 2 + 1 .
They may be infinite in number ; 48 are known as of 2013-02-05 .
p
2 - 1 is prime ===> p is prime
No odd perfect is known .
p
In binary radix representation , 2 - 1 is a repunit ,
e.g. , 11 , 111 , 11111 , 1111111 .
In radix-2 , every even perfect is a string of p-1 0s appended to a
string of p 1s :
e.g. , 110 , 11100 , 111110000 , 1111111000000 .
The sum of the reciprocals of the divisors > 1 of an even perfect is 1 ;
e.g. , 1/2 + 1/3 + 1/6 = 1 , 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 1 .
P mod 10 = 6 or 8
Let P be even perfect > 6 :
P mod 9 = 1
P mod 12 = 4
P mod 6 = 4
P mod 3 = 1
(p-1)/2
2
____
p-1 p \ 3
P = 2 ( 2 - 1 ) = > ( 2i - 1 )
/___
1
3 3 3 3
e.g. , 496 = 1 + 3 + 5 + 7
Elementary functions of even perfects
sum of divisors (P) Euler's phi (P)
# divisors (P) abundancy (P)
p p p-2 p
2p 2 ( 2 - 1 ) 2 ( 2 - 2 ) 2
---
Perfect Naturals
are not
Perfect Pow-ers ( e.g. , perfect squa-res , perfect cu-bes , etc. )
---
Odd Perfect Natural = OPN (??)
q[1] q[2] q[r]
p[1] * p[2] * ... * p[r] = OPN , where the p[i] are distinct primes
p[r] >= p[i]
p[1] mod 4 = 1
q[1] mod 4 = 1
For i > 1 , q[i] mod 2 = 0
OPN mod 36 = 1 or 9 or 13 or 25
300
OPN > 10
r
4
2 > OPN
r > 8
p[r] > 100000000
q[i] 20
p[i] > 10 , for some i
p[r-1] > 10000
p[r-2] > 100
____
\
> q[i] > 74
/___
r >= p[1]
---
Mersenne p p-1 p
rank p prime 2 - 1 Perfect 2 ( 2 - 1 )
1 2 3 6
2 3 7 28
3 5 31 496
4 7 127 8128
5 13 8191 33550336
6 17 131071 8589869056
7 19 524287 137438691328
8 31 2147483647 2305843008139952128
9 61 2305843009213693951 2658455991569831744654692615953842176
10 89
11 107 . .
12 127 . .
13 521 . .
14 607
15 1279
16 2203
17 2281
18 3217
19 4253
20 4423
21 9689
22 9941
23 11213
24 19937
25 21701
26 23209
27 44497
28 86243
29 110503
30 132049
31 216091
32 756839
33 859433
34 1257787
35 1398269
36 2976221
37 3021377
38 6972593
39 13466917
40 20996011
41 24036583
42 25964951
? 30402457
? 32582657
? 37156667
? 42643801
? 43112609
?? 57885161
By orders of magnitude = # of digits :
1 2 3 4 5 6 7 8
4 6 4 8 6 5 5 10+
---
metas :
elementary number theory
multiplicative number theory
additive number theory
sigma
abundancy
integer
natural number
sum of proper divisors
Mersenne prime
perfect number
perfect numbers
perfect integer
perfect integers
perfects
2-perfects
nombre parfait
nombres parfaits
entier parfait
entiers parfaits
vollkommene Zahl
vollkommene Zahlen
vollkommenen Zahlen
vollkommener Zahlen
perfekte Zahl
perfekte Zahlen
vollkommene Ganzzahl
vollkommene Ganzzahlen
perfekte Ganzzahl
perfekte Ganzzahlen
perfekt tal
perfekte tal
fuldkomne tal
n'umero perfecto
numero perfecto
n'umeros perfectos
numeros perfectos
entero perfecto
enteros perfectos
numerus perfectus
numeri perfecti
aliquot divisors
vollkommen
vollst"andig
vollstandig
deficient
diminute
defective
unvollkommen
unvollst"andig
unvollstandig
mangelhaft
abundant
superfluos
plus quam-perfectus
redundantem
exc'edant
excedant
"ubervollst"andig
ubervollstandig
"uberflussig
uberflussig
"uberschiessende
uberschiessende
tochnyi chislo (??)
tochnyi chisel (??)
---
Mathematicians and science writers , please take note :
+ +
For the integers , Z = J , the terms "divisor" and "factor" are not
interchangable as they are in ring theory
( more precisely , ideal theory ) .
Factors form a product . Free-standing integers are merely divisors .
These terms are well-formed :
divisor , factor , divisors , factors , the greatest common divisor ,
the set of divisors , the set of positive divisors , the set of proper
divisors , sum of the proper divisors , the set of prime factors ,
greatest common prime factor , the unique set of prime factors , a set
of factors , the sum of all its proper divisors .
These terms are ill-formed :
the set of factors , sum of the factors , greatest common factor .
These terms are incorrect :
the sum of all the divisors , the sum of all its factors .
E.g. , if the factors of 28 are purportedly 1 , 2 , 4 , 7 , and 14 ,
then 1 * 2 * 4 * 7 * 14 = 784 , thus exhibiting 1 , 2 , 4 , 7 , and 14
as a set of factors of 784 .
---
It takes no special mathematical nor computing skills to find the next
perfect number .
2010 Mathematics Subject Classification : Primary 11A25 ; Secondary 11A05 , 11A41 , 11A51 , 11A63 , 11N05 , 11N25 , 11Y05 , 11Y11 , 11Y55 , 11Y70
Walter Nissen
posted 2008-03-05 updated 2013-02-05