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A perfect number is a positive integer. In number theory, a perfect number is equal to the sum of its positive factors excluding the number itself. The smallest perfect number is 6. It is equal to the sum of 1,2 and 3.
In this article, we will discuss Perfect numbers, their definition, the history of perfect numbers and more.
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What are Perfect Numbers?
A perfect integer is a positive integer that can be expressed as the sum of its positive factors except for the number itself. In other words, perfect numbers are the positive integers that are the sum of their divisors. In mathematics, 6 is the smallest perfect number. Number 6 can be written as the sum of its factors: 1, 2, and 3.
Note: the factors used for the summation do not include the number itself.
What are Complete Numbers?
The sum of all the divisors of a number is equal to twice the number, such numbers are called complete numbers. One must note that all the perfect numbers are also complete numbers.
For example: factors of 6: 1, 2, 3, and 6
6 as perfect number: 1 + 2 + 3 = 6
6 as a complete number: 1 + 2 + 3 + 6 = 12
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History of Perfect Numbers
The concept of perfect numbers might date back to ancient Egypt, where these numbers could have been encountered naturally. The Information regarding the discovery of the Perfect number is not known.
This number got much attention from the Pythagoreans. It is a little of a mystery as to when Perfect Numbers were first discovered, by how they were studied, it is speculated that they may even have been of Egyptian origin, and may even have been known before (Burton, 1980).
Although the ancient mathematicians knew of the existence of Perfect Numbers, it was the Greeks who took a keen interest in them, especially Pythagoras and his followers, for example, the scholars O’Connor and Robertson (2004) write that every fifth number in the sequence of natural numbers, namely 6, 28, 496, and 8128 is perfect, that is it is equal to the sum of its proper divisors. Further, they imply that the Greeks did not know of the existence of them (O’Connor and Robertson, 2004).
The smallest perfect number is 6, which caught the attention of the Pythagoreans. They valued it not just for its mathematical properties but also for its mystical and numerological significance. Following 6, the next perfect number discovered was 28. These early observations marked the beginning of the exploration of perfect numbers in mathematics, blending early numerology with the roots of mathematical theory.
How to Find a Perfect Number?
To find a perfect number, you can use a technique used by Euclid. According to Euclid, an expression that can be a perfect number subject to a specific condition. According to his proposition, if 2n – 1 is a prime number then 2n-1(2n – 1) is a perfect number. Euclid said that 2n – 1 multiplied by 2n-1, is a perfect number. The term in the bracket that is (2n-1) is a prime number. In other words [2n-1 × (2n-1) = perfect number], where (2n-1) is a prime number.
| n | (2n-1) | (2n-1) | (2n-1) × (2n – 1) |
| 1 | 1 | 1 | – |
| 2 | 2 | 3 (prime number) | 6 (perfect number) |
| 3 | 4 | 7 (prime number) | 28 (perfect number) |
| 4 | 8 | 15 | – |
| 5 | 16 | 31 (prime number) | 496 (perfect number) |
| 6 | 32 | 63 | – |
| 7 | 64 | 127 (prime number) | 8128 (perfect number) |
| 8 | 128 | 255 | – |
| 9 | 256 | 511 | – |
| 10 | 512 | 1023 | – |
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Steps to Identify Perfect Numbers Using Expressions
Step 1: Start by substituting n=1 into the expressions 2n−1 and (2 n −1). Substituting n=1 into the first expression gives 21−1 = 1. Substituting n=1 into the second expression gives (21 − 1) = 1.
Step 2: Continue substituting values of n=2,3,4, and so on, and record the results in a table.
Step 3: Observe the column containing (2n −1). If this number is prime, then the product of the two expressions, 2n−1 and (2n −1), will result in a perfect number.
Step 4: For example, when n=2, the first expression 22 – 1 gives 3, and the second expression (22 −1) gives 3, both of which are prime. Similarly, when n=3, the first expression gives 5, and the second expression gives 7.
Step 5: By examining the table, notice that the prime numbers under the column (2 n −1) correspond to perfect numbers when multiplied by their respective 2n−1 values. For example, prime numbers such as 3, 7, 31, and 127 lead to perfect numbers 6, 28, 496, and 8128, respectively. This demonstrates that when (2n −1) is prime, the product of 2n−1 and (2n−1) will always yield a perfect number.
A few of the perfect numbers that are known throughout time are 6,28, etc. A list of a few more is provided below.
- 6 (1 + 2 + 3)
- 28 (1 + 2 + 4 + 7 + 14)
- 496 (1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248)
- 8128 (1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064)
Important Notes on Perfect Number
- Perfect numbers are those which are both a positive integer and the sum of its proper divisors, excluding itself.
- 6 is the smallest perfect number.
- All perfect numbers are even.
- Perfect numbers all end in 6 and 8 every other time.
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Solved Example on Perfect Numbers
Example 1: Check whether ’28’ is a perfect number.
Solution: Factors of 28 = 1, 2, 4, 7, 14, 28
The sum of factors excluding 28 is 1 + 2 + 4 + 7 + 14 = 28
Here the sum of the factors of 28 excluding 28 is 28 itself.
Therefore 28 is a perfect number.
Example 2: Check whether ‘6’ is a perfect number.
Solution: Factors of 6 = 1, 2, 3, 6
The sum of factors excluding 6 is 1 + 2 + 3 = 6.
Here the sum of the factors of 6 excluding is 6.
Therefore 6 is a perfect number.
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Practice Questions on Perfect Number
1. Check whether ’62’ is a perfect number.
2. Check whether ‘986’ is a perfect number.
3. Check whether ‘697’ is a perfect number.
4. Check whether ’23’ is a perfect number.
5. Check whether ‘245’ is a perfect number.
Perfect Number FAQs
What are the first 5 perfect numbers?
The first 5 perfect numbers are 6, 28, 496, 8128, and 33550336.
What are the largest perfect numbers?
The largest perfect number is unknown.
Is 9 a perfect number?
No, nine is not a perfect number.
