PERFECT NUMBERS - DEFINITION, FORMULA FOR GENERATING THEM, EXAMPLES, PROPERTIES

Please study  Mersenne Primes before Perfect Numbers,
if you have not already done so.

We need the knowledge of Mersenne Primes here.

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.

Factors of 8128
= { 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128 }
Proper Factors of 8128
= { 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 }
Sum of the proper factors of 8128
= 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064
= 8128.

In each of the four numbers 6, 28, 496 and 8128, we have seen that the sum of their proper factors is equal to the number.

These four perfect numbers 6, 28, 496 and 8128
have been known from ancient times.

If we look at the factors of each of these 4 numbers,
we can see a prime number (other than 2) in them.

3 for 6, 7 for 28, 31 for 496 and 127 for 8128.

We can also see an even factor preceding this prime factor.

2 before 3 for 6, 4 before 7 for 28, 16 before 31 for 496
and 64 before 127 for 8128.

We can also observe that the multiplication of
this even factor and the prime factor gives the number.

2 x 3 = 6; 4 x 7 = 28; 16 x 31 = 496; 64 x 127 = 8128.

We may also observe that
3 = 4 - 1 = 2² - 1; 7 = 8 - 1 = 2³ - 1;
31 = 32 - 1 = 2⁵ - 1; 127 = 32 - 1 = 2⁷ - 1;

We may also see that the even factor
preceding the prime factor is also a power of 2.

2 = 2¹; 4 = 2²; 16 = 2⁴; 64 = 2⁶.

Thus these numbers can be expressed as

6 = 2¹(2² - 1);
28 = 2²(2³ - 1);
496 = 2⁴(2⁵ - 1);
8128 = 2⁶(2⁷ - 1);

i.e. These four are of the form
2^{(n - 1)}(2ⁿ - 1), where n and (2ⁿ - 1) are Prime Numbers.

This is an interesting and useful observation.

Not only these four, but also
every even perfect number has this form.

Every even Perfect Number is of the form 2^{(n - 1)}(2ⁿ - 1)
where n and (2ⁿ - 1) are Prime Numbers.

Also,

If (2ⁿ - 1) is a Prime Number, then [2^{(n - 1)}][2ⁿ - 1] is an even Perfect Number.

The above two statements are called Euclid-Euler Formulas.

Relation Between Mersenne Primes and Perfect Numbers

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

If (2ⁿ - 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.

List of Mersenne Primes and Perfect Numbers

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.

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Properties of Perfect Numbers

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

2^{(n - 1)}(2ⁿ - 1) = (1⁄2)2ⁿ(2ⁿ - 1)

Let (2ⁿ - 1) = m.

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

Examples :

6 = 1 + 2 + 3
28 = 1 + 2 + 3 + 4 + 5 + 6 + 7
496 = 1 + 2 + 3 + .............+ 31
8128 = 1 + 2 + 3 + .............+ 127

Property 2 : any even perfect number except the first one is the sum of the first 2^(n−1)⁄2 odd cubes.

Examples :

28 = 1³ + 3³
496 = 1³ + 3³ + 5³ + 7³
8128 = 1³ + 3³ + 5³ + 7³ + 9³ + 11³ + 13³ + 15³

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.

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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 10³⁰⁰.
That means there is no odd perfect number below 10³⁰⁰.

(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 10⁸.

(ix) should have the second largest prime factor greater than 10⁴.

(x) should have the third largest prime factor greater than 10².