We know the factors or divisors of a natural number are the ones which divide the number exactly without remainder.

If the number itself is excluded from the factors (divisors), the factors (divisors) are called proper factors (divisors).

Comparing the sum of the proper factors (divisors) with the number was of interest in Greek numerology.

Depending on the number being greater than or equal to or smaller than the sum of its proper factors (divisors), they called the number abundant or perfect or deficient respectively.

Later, the study of the numbers which are equal to the sum of their proper factors (divisors) became an important topic for pure mathematicians in number theory.

Definition of Perfect Numbers

A natural number is said to be a Perfect Number, if the sum of its proper factors is equal to the number.

Examples :

Factors of 6 = { 1, 2, 3, 6 } Proper Factors of 6 = { 1, 2, 3 } Sum of the proper factors of 6 = 1 + 2 + 3 = 6.

Factors of 28 = { 1, 2, 4, 7, 14, 28 } Proper Factors of 28 = { 1, 2, 4, 7, 14 } Sum of the proper factors of 28 = 1 + 2 + 4 + 7 + 14 = 28.

Factors of 496 = { 1, 2, 4, 8, 16, 31, 62, 124, 248, 496 } Proper Factors of 496 = { 1, 2, 4, 8, 16, 31, 62, 124, 248 } Sum of the proper factors of 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496.

Relation Between Mersenne Primes and Perfect Numbers

We may recall that the prime numbers which are of the form (2^{n} - 1) are called
Mersenne Primes.
So,

If (2^{n} - 1) is a Prime, it is a Mersenne Prime. And even Perfect Number = 2^{(n - 1)} x that Mersenne Prime.

∴ Corresponding to every Mersenne Prime, there is an even Perfect Number, and there is one-one relation between them.

We know, there are 44 known Mersenne Primes, as of August 2008 and hence there are 44 known even Perfect Numbers.

Out of these 44 Mersenne primes, the first 39 are clear. That means there are no other Mersenne primes in between any two of these 39.

But, it is not known whether there exist any Mersenne Primes in between any two of the discovered 39th to 44th Mersenne Primes.

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In the following Table, the values of the exponent n, the values of or the number of digits in the Mersenne primes for all the 44 ones are given.

The values of or the number of digits in the Perfect Numbers are given for the first 10 numbers and the 44th. For the remaining, you may consider the number of digits to be nearly equal to double that of Mersenne Primes.

S.No.

n

Mersenne Prime = (2^{n} - 1)

Perfect Number = 2^{(n - 1)}(2^{n} - 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

Has 19 digits

Has 37 digits

10

89

Has 27 digits

Has 54 digits

11

107

Has 33 digits

12

127

Has 39 digits

13

521

Has 157 digits

14

607

Has 183 digits

15

1,279

Has 386 digits

16

2,203

Has 664 digits

17

2,281

Has 687 digits

18

3,217

Has 969 digits

19

4,253

Has 1,281digits

20

4,423

Has 1,332 digits

21

9,689

Has 2,917 digits

22

9,941

Has 2,993 digits

23

11,213

Has 3,376 digits

24

19,937

Has 6,002 digits

25

21,701

Has 6,533 digits

26

23,209

Has 6,987 digits

27

44,497

Has 13,395 digits

28

86,243

Has 25,962 digits

29

110,503

Has 33,265 digits

30

132,049

Has 39,751 digits

31

216,091

Has 65,050 digits

32

756,839

Has 227,832 digits

33

859,433

Has 258,716 digits

34

1,257,787

Has 378,632 digits

35

1,398,269

Has 420,921 digits

36

2,976,221

Has 895,932 digits

37

3,021,377

Has 909,526 digits

38

6,972,593

Has 2,098,960 digits

39

13,466,917

Has 4,053,946 digits

40

20,996,011

Has 6,320,430 digits

41

24,036,583

Has 7,235,733 digits

42

25,964,951

Has 7,816,230 digits

43

30,402,457

Has 9,152,052 digits

44

32,582,657

Has 9,808,358 digits

Has 19,616,714 digits

Properties of Perfect Numbers

Property 1 : Any even perfect number is the sum of the first (2^{n} - 1) natural numbers and so is a Triangular Number.

2^{(n - 1)}(2^{n} - 1) = (1⁄2)2^{n}(2^{n} - 1)

Let (2^{n} - 1) = m.

Then, 2^{(n - 1)}(2^{n} - 1) = m(m + 1)⁄2 which is a formula for the sum of the first m (= 2^{n} - 1) natural numbers.

We know, the number which is equal to the sum of natural numbers upto some number is called Triangular Number.

∴ every even Perfect Number is a Triangular Number.

Property 3 : any even perfect number except the first one gives the remainder 1 when divided by 9.

You may check this property by applying
Divisibility Rule
for 9. If the sum of the digits of the number is divisible by 9 (gives remainder 0), then the number is divisible by 9.

A slight modification of the above rule is given below.

If dividing the sum of the digits of a number by 9 gives remainder 1, then the number gives remainder 1 when divided by 9.

Examples :

sum of the digits of 28 = 2 + 8 = 10 gives remainder 1 when divided by 1.

sum of the digits of 496 = 4 + 9 + 6 = 19 gives remainder 1 when divided by 1.

sum of the digits of 8128 = 8 + 1 + 2 + 8 = 19 gives remainder 1 when divided by 1.

Hence, the perfect numbers 28, 496, 8128 give remainder 1 when divided by 9.

Property 4 : The sum of the reciprocals of all the factors including itself, of a perfect number equals to 2.

Examples :

Factors of 6 = { 1, 2, 3, 6 } ∴ The sum of the reciprocals of all the factors = 1⁄1 + 1⁄2 + 1⁄3 + 1⁄6 = (6 + 3 + 2 + 1)⁄6 = 12⁄6 = 2.

Factors of 28 = { 1, 2, 4, 7, 14, 28} ∴ The sum of the reciprocals of all the factors = 1⁄1 + 1⁄2 + 1⁄4 + 1⁄7 + 1⁄14 + 1⁄28 = (28 + 14 + 7 + 4 + 2 + 1)⁄28 = 56⁄28 = 2.

Odd Perfect Numbers

Till date (August 2008), there is no odd Perfect Number discovered.

The question of the existence or otherwise of odd Perfect Numbers is still an unsolved question in Number Theory.

The research carried out till date gave many points.

Some of them are

Odd Perfect Number, if it exists,

(i) should be greater than 10^{300}. That means there is no odd perfect number below 10^{300}.

(ii) is not divisible by 105.

(iii) is of the form 12m + 1 or 36m + 9.

(iv) should have more than or equal to 75 prime factors.

(vi) should have more than or equal to 9 distinct prime factors.

(vii) should have more than or equal to 12 distinct prime factors, if 3 is not a factor of it.

(viii) should have the largest prime factor greater than 10^{8}.

(ix) should have the second largest prime factor greater than 10^{4}.

(x) should have the third largest prime factor greater than 10^{2}.

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