A perfect
number is a number which is equal to the sum of all its divisors less than
itself. For example, 6 is a perfect
number because all of its divisors less than itself are 1, 2, and 3; and 1 + 2
+ 3 = 6. The number 28 is another
perfect number because all of its divisors less than itself are 1, 2, 4, 7, and
14; and 1 + 2 + 4 + 7 + 14 = 28.
Another way
to define a perfect numbers is to state that is a number whose sum of all of
its factors (including itself) is equal to twice its value. For example, 6 is a perfect number because
all of its factors are 1, 2, 3, and 6; and 1 + 2 + 3 + 6 = 12, which is twice
its value. The number 28 is another perfect
number because all of its factors are 1, 2, 4, 7, 14, and 28; and 1 + 2 + 4 + 7
+ 14 + 28 = 56, which is also twice its value.
A visual
representation of the perfect number 6
A computer
can be used to quickly find other perfect numbers. The following program goes through every integer
(less than a user-defined integer), finds its factors, adds them up, and
displays the ones that are perfect along with its factors:
01
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# Perfect Numbers
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02
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# Python 2.7.3
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03
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# To find all perfect numbers less than n, type
"perfects(n)". For example,
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04
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# to find
all perfect numbers less than 1000, type "perfects(1000)".
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05
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06
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# prints a string with a hanging indentation
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07
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import textwrap
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08
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def hprint(string):
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09
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print textwrap.fill(textwrap.dedent(
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10
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string).strip(), initial_indent =
"", subsequent_indent = "
")
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11
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12
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# returns a list of factors of n
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13
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def factor(n):
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14
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arrFactors = []
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15
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# go
through all the numbers < or = the square root of n
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16
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for i in range(1, int(n ** 0.5) + 1):
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17
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#
if the numbers divides into n, add it to the list
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18
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if (n % i == 0):
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19
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arrFactors.append(i)
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20
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# it is different, add the other factor to the list, too
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21
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if (i != n / i):
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22
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arrFactors.append(n / i)
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23
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#
return the sorted list
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24
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return sorted(arrFactors)
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25
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26
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# returns the sum of all the factors of n
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27
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def o(n):
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28
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intO = 0
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29
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# get
the factors of n
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30
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arrFactors = factor(n)
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31
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# go
through all the factors and add them up
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32
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for i in range(0, len(arrFactors)):
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33
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intO += arrFactors[i]
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34
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#
return the sum
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35
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return intO
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36
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37
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# determines if n is a perfect number
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38
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def perfect(n):
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39
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# if
sigma of n (the sum of the factors) equals 2n, then it is a
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40
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# perfect number, so display it and its
factors, and return True
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41
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if (o(n) == 2 * n):
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42
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hprint(str(n) + ": " +
str(factor(n)))
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43
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return True
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44
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# if
not, return False
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45
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else:
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46
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return False
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47
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48
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# find all the perfect numbers less than n
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49
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def perfects(n):
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50
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# go
through all the numbers between 1 and n and test them for being
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51
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# a perfect number
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52
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for i in range(1, n):
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53
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perfect(i)
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Typing in
“perfects(10000)” results in all the perfect numbers less than 10,000, and
there are only 4: 6, 28, 496, and 8128.
>> perfects(10000)
6: [1, 2, 3, 6]
28: [1, 2, 4, 7, 14, 28]
496: [1, 2, 4, 8, 16, 31, 62,
124, 248, 496]
8128: [1, 2, 4, 8, 16, 32, 64,
127, 254, 508, 1016, 2032, 4064, 8128]
Amazingly, the
Ancient Greeks knew about all 4 of these perfect numbers in an age without
computers. In fact, around 300 B.C.
Euclid proved that if 2
p – 1 is prime, then 2
p – 1(2
p
– 1) is a perfect number. For example, 3
is a prime number for p = 2 (2
p – 1
= 2
2 – 1
= 3), so 2
p – 1(2
p – 1) = 2
2 – 1(2
2
– 1) = 2(3) = 6 is a perfect number. Also,
7 is a prime number for p = 3 (2
p – 1
= 2
3 – 1
= 7), so 2
p – 1(2
p – 1) = 2
3 – 1(2
3
– 1) = 4(7) = 28 is a perfect number.
Euclid’s proof can be found
here (
Proof PNP01).
Euclid
Since many
perfect numbers are in the form 2p – 1(2p – 1), the
following functions can be added to improve the above computer program to test
2p – 1(2p – 1) as a perfect number for any given p value:
55
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# determines if (2 ** (p - 1)) * (2 ** p - 1) is
a perfect number
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56
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def pperfect(p):
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57
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if perfect((2 ** (p - 1)) * (2 ** p - 1))
== False:
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58
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return False
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59
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|
60
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# find all the perfect numbers in the form of (2
** (p - 1)) * (2 ** p - 1)
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61
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# less
than the given value of p
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62
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def pperfects(p):
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63
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# go
through all the numbers between 1 and p and test
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64
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# (2 ** (p - 1)) * (2 ** p - 1) for being a
perfect number
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65
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for i in range(1, p):
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66
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pperfect(i)
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Typing in
“pperfects(25)” quickly gives 3 more perfect numbers 33,550,336 and 8,589,869,056
and 137,438,691,328:
>> pperfects(25)
6: [1, 2, 3, 6]
28: [1, 2, 4, 7, 14, 28]
496: [1, 2, 4, 8, 16, 31, 62,
124, 248, 496]
8128: [1, 2, 4, 8, 16, 32, 64,
127, 254, 508, 1016, 2032, 4064, 8128]
33550336: [1, 2, 4, 8, 16, 32,
64, 128, 256, 512, 1024, 2048, 4096,
8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896,
4193792, 8387584, 16775168, 33550336]
8589869056: [1, 2, 4, 8, 16, 32,
64, 128, 256, 512, 1024, 2048, 4096,
8192, 16384, 32768, 65536, 131071L, 262142L, 524284L, 1048568L,
2097136L, 4194272L, 8388544L, 16777088L, 33554176L, 67108352L,
134216704L, 268433408L, 536866816L, 1073733632L, 2147467264L,
4294934528L, 8589869056L]
137438691328: [1, 2, 4, 8, 16,
32, 64, 128, 256, 512, 1024, 2048,
4096, 8192, 16384, 32768, 65536, 131072, 262144, 524287L, 1048574L,
2097148L, 4194296L, 8388592L, 16777184L, 33554368L, 67108736L,
134217472L, 268434944L, 536869888L, 1073739776L, 2147479552L,
4294959104L, 8589918208L, 17179836416L, 34359672832L, 68719345664L,
137438691328L]
Unfortunately,
testing p ≥
29 results in a “MemoryError” on my current personal computer, but
supercomputers can be used to find more perfect numbers.
Euclid’s
proof showed that perfect numbers are directly related to prime numbers in the
form of 2
p – 1, and these prime numbers later became known as
Mersenne primes, named after the French mathematician who studied them extensively
in the 17
th century. Over the
years since, there have been many clever tricks developed to find larger and
larger Mersenne prime numbers with the help of computers (see
here for more
details), and currently the largest known Mersenne prime is 2
74,207,281
– 1, a number with 22,338,618 digits!
According to Euclid’s proof, since 2
74,207,281 – 1 is a prime
number, 2
74,207,280(2
74,207,281 – 1) must be a perfect
number, currently making it the largest known perfect number, a number with 44,677,235
digits!
Mersenne
Interesting Facts
Here are
some more interesting facts about perfect numbers:
-
Euler proved that all even perfect numbers must
be in the form of 2
p – 1(2
p – 1) (
Proof PNP02).
Euler
-
There are no known odd perfect numbers. Computers have verified that there are no odd
perfect numbers below 101500.
-
It is not known if there is an infinite number
of perfect numbers. (It is also not
known if there is an infinite number of Mersenne primes. Proving one would prove the other.)
-
All even perfect numbers (in the form of 2
p
– 1(2
p – 1)) can be represented in binary as p ones followed
by p − 1 zeros (
Proof PNP03):
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perf
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binary
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6
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110
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28
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11100
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496
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111110000
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8128
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11111111000000
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All even perfect numbers (in the form of 2
p
– 1(2
p – 1))are the sum of all the integers from 1 to 2
p – 1 (also called 2
p
– 1 triangular numbers) (
Proof PNP04):
|
perf
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Sum
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6
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1 + 2 + 3
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28
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1 + 2 + 3 + 4 + 5 + 6 + 7
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496
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1 + 2 + 3 + … + 29 + 30 + 31
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8128
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1 + 2 + 3 + … + 125 + 126 + 127
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All even perfect numbers except for 6 (in the
form of 2
p – 1(2
p – 1)) are the sum of the first 2
(p−1)/2
odd cubes (
Proof PNP05):
|
perf
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Sum
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28
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13 + 33
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496
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13 + 33 + 53
+ 73
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8128
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13 + 33 + 53
+ 73 + 93 + 113 + 133 + 153
|
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All even perfect numbers except for 6 (in the
form of 2
p – 1(2
p – 1)) are in the form of 1 + 9(T
(2^p-2)/3),
where T
(2^p-2)/3 is the sum of all the integers from 1 to
2^p –
2 /
3 (also called
2^p – 2 /
3
triangular numbers) (
Proof PNP06):
|
perf
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sum
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28
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1 + 9(1 + 2)
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496
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1 + 9(1 + 2 + 3 + … + 8 + 9 + 10)
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8128
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1 + 9(1 + 2 + 3 + … + 40 + 41 + 42)
|
-
All even perfect numbers except 6 have a digital
root of 1 (to find a digital root of a number, add the digits, then add the
digits of the resulting number, then repeat the process until there is a single
digit) (
Proof PNP07):
|
perf
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digit
sum
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28
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2 + 8 = 10, 1 + 0 = 1
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496
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4 + 9 + 6 = 19, 1 + 9 = 10, 1 + 0 = 1
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8128
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8 + 1 + 2 + 8 = 19, 1 + 9 = 10, 1 + 0
= 1
|
-
The sum of reciprocals of all the divisors of
every perfect number is equal to 2 (
Proof PNP08)
|
perf
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reciprocal
sum
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6
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1/1 + 1/2
+ 1/3 + 1/6 = 2
|
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28
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1/1 + 1/2
+ 1/4 + 1/7 + 1/14
+ 1/28 = 2
|
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496
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1/1 + 1/2
+ 1/4 + … + 1/248 + 1/496
= 2
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8128
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1/1 + 1/2
+ 1/4 + … + 1/4064 + 1/8128
= 2
|
-
Any number raised to a perfect number exponent
is equal to the product of all its perfect roots. For example, 6 is a perfect number, 2
6
= 64, the perfect roots of 64 are
6√64 = 2,
3√64
= 4, and √64
= 8, and the product of 2, 4, and 8 is 2 ∙ 4 ∙ 8 = 64 (
Proof PNP09).
|
perf
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root
product
|
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6
|
6√x6 ∙ 3√x6
∙
2√x6 = x6
|
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28
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28√x28∙14√x28∙7√x28∙4√x28∙2√x28
= x28
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496
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496√x496∙248√x496∙…∙4√x496∙2√x496
= x496
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8128
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8128√x8128∙4064√x8128∙…∙4√x8128∙2√x8128
= x8128
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All even perfect numbers end in 6 or 8 (
Proof PNP10). All even perfect numbers with
more than 1 digit end in 16, 28, 36, 56, 76, or 96 (
Proof PNP11).
|
perf
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last digit
|
last 2 digits
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6
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6
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n/a
|
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28
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8
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28
|
|
496
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6
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96
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8128
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8
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28
|
Related Concepts
Like perfect
numbers, there are many other types of numbers defined by the sum of their
divisors:
-
In the book Cutting
for Stone by Abraham Verghese, one of the characters explains about a set
of numbers whose sum of divisors is a perfect square:
“’Sixty-six is my second-favorite number,’
Shiva said. ‘Pray, why is it your second favorite?’ said Bailey. ‘Because if
you take the numbers you can divide into sixty-six, including sixty-six, and
add them up, what you have is a square.’ Mr. Bailey couldn't resist. He wrote
down 1, 2, 3, 6, 11, 22, 33, and 66 – all the numbers that went into 66 – and
then he totaled. What he got was 144, at which point both he and Shiva said,
‘Twelve squared!’ ‘That's what makes sixty-six special,’ Shiva said. ‘It's also
true of three, twenty-two, sixty-six, seventy – their divisors add up to a
square.’”
Cutting for Stone
-
An
abundant number is a number in which the sum of all its divisors less than
itself is greater than the number
itself. For example, 12 is an abundant
number because its divisors less than itself are 1, 2, 3, 4, and 6; and 1 + 2 +
3 + 4 + 6 = 16 which is greater than 12.
-
A
deficient number is a number in which the sum of all its divisors less than
itself is less than the number
itself. For example, 8 is a deficient
number because its divisors less than itself are 1, 2, and 4; and 1 + 2 + 4 = 7
which is less than 8.
-
Amicable
numbers are a pair of numbers in which the sum of all the divisors less
than itself of the first number is equal to the second number, and the sum of
all the divisors less than itself of the second number is equal to the first
number. For example, 220 and 284 are
amicable numbers because all of the factors less than itself of 220 are 1, 2,
4, 5, 10, 11, 20, 22, 44, 55, and 110; and 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 +
110 = 284; and all the factors less than itself of 284 are 1, 2, 4, 71, and
142; and 1 + 2 + 4 + 71 + 142 = 220.
Other amicable numbers include 1184 and 1210, 2620 and 2924, 5020 and
5564, 6232 and 6368, and many more.
Conclusion
Perfect
numbers have a simple definition but a wealth of related facts. Yet despite the plethora of postulates
concerning perfect numbers, there is still much to prove, including whether or
not an odd perfect number exists, and whether or not there is an infinite
number of perfect numbers. Hopefully
these mysteries will inspire mathematical works in the future.
