### Concise , remarkable facts about perfect numbers

```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
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 .
Check out the incredible GIMPS , the Great Internet Mersenne Prime Search , a most advanced part of the search for 2-perfect naturals .
http://www.mersenne.org

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
```