Perfect numbers how to identify them and examples

A perfect number is a natural number such that the sum of its divisors is the same as the number. Obviously the number itself cannot be included between the divisors.

One of the simplest examples of a perfect number is 6, since its divisors are: 1, 2 and 3. If we add the divisors, we obtain: 1 + 2 + 3 = 6.

The sum of the divisors of an integer, not including the number itself, is called aliquot. Therefore a perfect number is equal to its aliquot.

But if the number itself is included in the sum of divisors of a number, then a perfect number will be one that the sum of all its divisors divided by 2 is equal to the number itself..

Article index

• 1 History
• 2 Properties of perfect numbers
  • 2.1 Euclid's formula and criterion
  • 2.2 The largest known perfect number
  • 2.3 A perfect number is friends with itself
• 3 Examples of perfect numbers
• 4 Exercises
  • 4.1 - Exercise 1
  • 4.2 - Exercise 2
  • 4.3 - Exercise 3
  • 4.4 - Exercise 4
• 5 References

Story

The mathematicians of antiquity, particularly the Greeks, attached great importance to perfect numbers and attributed divine qualities to them..

For example, Philo of Alexandria, towards the first century, affirmed that 6 and 28 are perfect numbers that coincide with the six days of the creation of the world and the twenty-eight days it takes for the Moon to go around the Earth..

Perfect numbers are also present in nature, for example at the north pole of Saturn the perfect number 6 also appears, a hexagon-shaped vortex found by the Cassini probe and that has scientists intrigued.. 

The honeycombs of bees have hexagonal shaped cells, that is, with 6 sides. It has been shown that the polygon with the perfect number 6 is the one that allows maximizing the number of cells in the bee hive, with the minimum of wax for its elaboration..

Properties of perfect numbers

The sum of all the divisors of a natural number n is denoted by σ (n). In a perfect number it is satisfied that: σ (n) = 2n.

Euclid's formula and criteria

Euclid discovered a formula and a criterion that allows finding the perfect numbers. This formula is:

two^(n-1) (twoⁿ -1)

However, the number generated by the formula will be perfect only when the factor (2ⁿ -1) be a cousin.

Let's see how the first perfect numbers are generated:

If n = 2 then we have 2 left¹ (two^two - 1) = 2 x 3 = 6 which we already saw is perfect.

When n = 3 we have 2^two (two³ - 1) = 4 x 7 = 28 which is also perfect as verified in detail in example 1.

Let's see what happens with n = 4. When substituting in Euclid's formula we have:

two³ (two⁴ - 1) = 8 x 15 = 120

It can be verified that this number is not perfect, as shown in detail in Example 3. This does not contradict Euclid's criterion, since 15 is not prime, a necessary requirement for the result to be a perfect number.

Let's now see what happens when n = 5. Applying the formula we have:

two⁴ (two⁵ - 1) = 16 x 31 = 496

Since 31 is a prime number, then the number 496 has to be perfect, according to Euclid's criteria. Example 4 shows in detail that it is indeed.

Prime numbers that have the form 2^p - 1 are called Mersenne cousins, after the monk Marin Mersenne, who studied prime numbers and perfect numbers back in the 17th century..

Later in the 18th century Leonhard Euler showed that all perfect numbers generated by Euclid's formula are even.

To date no perfect has been found that is odd.

The largest known perfect number

To date, 51 perfect numbers are known, all generated using Euclid's formula and criteria. This number was obtained once the larger Mersenne cousin was found, which is: (2^82589933 - 1).

The perfect number # 51 is (2^82589933) x (2^82589933 - 1) and has 49724095 digits.

A perfect number is friends with itself

In number theory it is said that two numbers are friends when the sum of the divisors of one, not including the number itself, is equal to the other number and vice versa.

The reader can verify that the sum of the divisors of 220, not including 220 is 284. On the other hand, the sum of the divisors of 284, not including 284, is equal to 220. Therefore the pair of numbers 220 and 284 are friends.

From this point of view, a perfect number is friends with itself..

Examples of perfect numbers

The first eight perfect numbers are listed below:

6

28

496

8128

33550336

8589869056

137438691328

2305843008139952128

Training

In the following exercises it will be necessary to calculate the divisors of a number, and then add them and verify if the number is a perfect number or not.

Therefore, before approaching the exercises, we will review the concept and show how they are calculated..

To begin with, you have to remember that numbers can be prime (when they can only be divided exactly with itself and 1) or composite (when they can be decomposed as a product of prime numbers).

For a composite number N we have:

N = aⁿ . b^m. c^p ... r^k 

Where a, b, c… r are prime numbers and n, m, p… k are exponents belonging to the natural numbers, which can be from 1 onwards.

In terms of these exponents, there is a formula to know how many divisors the number N has, although it does not tell us what these are. Let C be this quantity, then:

C = (n +1) (m + 1) (p +1)… (k + 1)

Decomposing the number N as a product of prime numbers and knowing how many divisors it has, both prime and non-prime, will help us determine what these divisors are..

Once you have all of them, except the last one that is not required in the sum, you can check if it is a perfect number or not.

- Exercise 1

Verify that the number 28 is perfect.

Solution

The first thing will be to decompose the number into its prime factors.

28 | 2
14 | 2
07 | 7
01 | 1

Its divisors are: 1, 2, 4, 7, 14 and 28. If we exclude 28, the sum of the divisors gives:

1 + 2 + 4 + 7 + 14 = 3 + 4 + 7 + 14 = 7 + 7 + 14 = 14 + 14 = 28

Therefore 28 is a perfect number.

Furthermore, the sum of all its divisors is 28 + 28 so the rule σ (28) = 2 x 28 is fulfilled.

- Exercise 2

Deciding if the number 38 is perfect or not.

Solution

The number is decomposed into its prime factors:

39 | 3
13 | 13
01 | 1

The divisors of 39 without including the number itself are: 1, 3 and 13. The sum 1 + 3 + 13 = 4 + 13 = 17 is not equal to 39, therefore 39 is an imperfect or non-perfect number. 

- Exercise 3

Find out if angel number 120 is perfect or imperfect.

Solution

We proceed to decompose the number into its prime factors:

120 | 2
060 | 2
 30 | 2
 15 | 3
  5 | 5
  1 | 1

From the prime factors we proceed to find the divisors:

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120

If 120 were perfect when adding all its divisors, we should obtain 2 x 120 = 240. 

1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360

This result is clearly different from 240, so it is concluded that the number 120 is not a perfect number.

- Exercise 4

Verify that the number 496, obtained by Euclid's criterion, is a perfect number.

Solution

The number 496 is decomposed into its prime factors:

496 | 2
248 | 2
124 | 2
062 | 2
031 | 31
001 | 1

So its divisors are:

1, 2, 4, 8, 16, 31, 62, 124, 248, 496

Now all of them are added, except 496:

1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

Confirming that it is indeed a perfect number.

References

1. Baldor, A. 1986. Arithmetic. Codex Editions and Distributions.
2. All about prime numbers. Friendly numbers. Recovered from: numeroprimos.org.
3. Wolfram MathWorld. Euler's Rule. Recovered from: mathworld.wolfram.com.
4. Wolfram MathWorld. Perfect number. Recovered from: mathworld.wolfram.com.
5. Wikipedia. Perfect numbers. Recovered from: en.wikipedia.org.
6. Wikipedia. Friendly numbers. Recovered from: es.wikipedia.org.