A perfect number is a positive integer equal to the total of its positive divisors, except the number itself in number theory. For example, 6 is a perfect number since 1 + 2 + 3 equals 6.
Some of the first perfect numbers are 6, 28, 496, and 8128. Perfect numbers are also known as “Complete Numbers” and “Proper Numbers“.
This article explores perfect numbers, covering their definition, examples, Euclid’s Perfect Number theorem, and methods to find them. We’ll also address FAQs and solve examples.
What is a Perfect Number?
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. Few of the starting Perfect Numbers are 6, 28, 496, 8128, and so on.
Perfect Number Definition
A positive integer that is equal to the sum of its positive divisors (excluding the number itself) is described as a Perfect Number.
Perfect Number Examples
Perfect Numbers range from the smallest, which is 6 up to infinity, few of the starting Perfect Numbers are
- 6
- 28
- 496
- 8128
- 130816
- 2096128
- 33550336 up to infinity.
The latest Perfect Number was discovered in 2018 has 49,724,095 digits.
Mersenne Prime Numbers
In mathematics, a Mersenne prime is a prime number that is one less than a power of two.
It’s represented as Mₙ = 2ⁿ − 1 for an integer n.
For instance, 31 is a Mersenne prime because it’s 2⁵ − 1.
The initial Mersenne primes include 3, 7, 31, and 127. 46th known Mersenne prime, discovered in 2008, is (237156667 − 1). Mersenne primes and perfect numbers are closely linked types of natural numbers in number theory.
Perfect Number Table
The table added below contains the starting 9 Mersenne Primes and their respected Perfect Numbers.
| Prime, (p) |
Mersenne Prime, (2p -1) |
Perfect Number, {2p-1(2p -1)} |
| 2 |
3 |
6 |
| 3 |
7 |
28 |
| 5 |
31 |
496 |
| 7 |
127 |
8128 |
| 13 |
8191 |
33550336 |
| 17 |
131071 |
8589869056 |
| 19 |
524287 |
137438691328 |
| 31 |
2147483647 |
2305843008139952128 |
| 61 |
2305843009213693951 |
2658455991569831744654692615953842176 |
History of Perfect Numbers
The history of perfect number is very old it goes back to Egytian civilization as they are one who first thought about Perfect Numbers. The major development in the study of perfect numbers is credited to Greeks, who eagerly read about Perfect numbers.
How to Find Perfect Numbers?
For example, let’s consider the number 6. Its divisors are 1, 2, and 3 (excluding 6). Adding these divisors gives 1 + 2 + 3 = 6. Therefore, by definition, 6 is a perfect number.
Another example is the number 496. Its divisors (factors) are 1, 2, 4, 8, 16, 31, 62, 124, and 248 (excluding 496). Adding these divisors results in 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496, which, according to the definition, is a perfect number.
Euclid’s Perfect Number Theorem
Euclid–Euler Theorem, also known as Euclid’s Perfect Number Theorem, connects Perfect Numbers to Mersenne Primes. It states that an even number is perfect if and only if it can be expressed in the form [2(p−1)(2p − 1)] where 2p-1 is a prime number.
Jacques Lefèvre, in 1496, suggested that the Euclid-Euler theorem encompasses all Perfect Numbers, implying the non-existence of odd Perfect Numbers.
According to Euclid’s Perfect Number theorem:
2p-1(2p-1) is an even perfect number where we have 2p-1 as a prime.
Similarly, we can generate the first four Perfect Number using the above formula (p is prime number):
p = 2: 21(22-1) = 2 × 3 = 6
p = 3: 22(23-1) = 4 × 7 = 28
p = 5: 24(25-1) = 16 × 31 = 496
p = 7: 26(27-1) = 64 × 127 = 8128
Perfect Number List
Few of the Perfect Number are 6, 28, 496, 8128, 130816, 2096128 the list goes on. Perfect number become less frequent as you go to higher values they have been studied in mathematics for centuries, and while an infinite number of them is not known, their properties and characteristics are well known.
|
Perfect Number
|
Sum of its Divisors
|
|
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
|
List of All 51 Perfect Numbers
Below is a list of all the 51 perfect numbers in ascending order:
| Serial Number |
Perfect Number |
Perfect Number Digits |
| 1 |
6 |
1 |
| 2 |
28 |
2 |
| 3 |
496 |
3 |
| 4 |
8128 |
4 |
| 5 |
33550336 |
8 |
| 6 |
8589869056 |
10 |
| 7 |
137438691328 |
12 |
| 8 |
230584…952128 |
19 |
| 9 |
265845…842176 |
37 |
| 10 |
191561…169216 |
54 |
| 11 |
131640…728128 |
65 |
| 12 |
144740…152128 |
77 |
| 13 |
235627…646976 |
314 |
| 14 |
141053…328128 |
366 |
| 15 |
541625…291328 |
770 |
| 16 |
108925…782528 |
1,327 |
| 17 |
994970…915776 |
1,373 |
| 18 |
335708…525056 |
1,937 |
| 19 |
182017…377536 |
2,561 |
| 20 |
407672…534528 |
2,663 |
| 21 |
114347…577216 |
5,834 |
| 22 |
598885…496576 |
5,985 |
| 23 |
395961…086336 |
6,751 |
| 24 |
931144…942656 |
12,003 |
| 25 |
100656…605376 |
13,066 |
| 26 |
811537…666816 |
13,973 |
| 27 |
365093…827456 |
26,790 |
| 28 |
144145…406528 |
51,924 |
| 29 |
136204…862528 |
66,530 |
| 30 |
131451…550016 |
79,502 |
| 31 |
278327…880128 |
130,100 |
| 32 |
151616…731328 |
455,663 |
| 33 |
838488…167936 |
517,430 |
| 34 |
849732…704128 |
757,263 |
| 35 |
331882…375616 |
841,842 |
| 36 |
194276…462976 |
1,791,864 |
| 37 |
811686…457856 |
1,819,050 |
| 38 |
955176…572736 |
4,197,919 |
| 39 |
427764…021056 |
8,107,892 |
| 40 |
793508…896128 |
12,640,858 |
| 41 |
448233…950528 |
14,471,465 |
| 42 |
746209…088128 |
15,632,458 |
| 43 |
497437…704256 |
18,304,103 |
| 44 |
775946…120256 |
19,616,714 |
| 45 |
204534…480128 |
22,370,543 |
| 46 |
144285…253376 |
25,674,127 |
| 47 |
500767…378816 |
25,956,377 |
| 48 |
169296…130176 |
34,850,340 |
| 49 |
451129…315776 |
44,677,235 |
| 50 |
109200…301056 |
46,498,850 |
| 51 |
110847…207936 |
49,724,095 |
These numbers follow the pattern [2p-1(2p -1)] where 2p−1 is a prime number.
Solved Questions on Perfect Numbers
Question 1: Is 28 a Perfect Number or not?
Solution:
Divisors of 28 are : 1, 2, 4, 7, 14 (excluding 28)
On adding the divisors,
1 + 2 + 4 + 7 + 14 =28
which hence proves that 28 is a Perfect Number.
Question 2: Is 56 a Perfect Number or not?
Solution:
Divisors of 56 are: 1, 2, 4, 7, 8, 14, 28 (excluding 56)
On adding the divisor,
1 + 2 + 4 + 7 + 8 + 14 + 28 = 64
which hence proves that 56 is not a Perfect Number.
Question 3: Is 8128 a Perfect Number or not?
Solution:
Divisors of 8128: 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 (excluding 8128)
On adding the divisors,
1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128
which hence proves that 8128 is a Perfect Number.
Practice Problems on Perfect Number
Q1. The sum of all of the reciprocals of a perfect number’s factors (including the perfect number itself) equals?
Q2. The first 4 perfect numbers can be generated by using the formula 2n-1 x [2n -1], where n = 2, 3, 5 . This formula was discovered by?
Q3. Notice the following pattern and answer: The first perfect number has one digit; the second perfect number has two digits; the third one has three digits; and the fourth one has four digits. So, does the fifth perfect number contain five digits?
Perfect Numbers – FAQs
What is Perfect Number definition?
The number obtained by adding its factors is equal to the number itself then the number is called as the perfect number.
What is the Smallest Perfect Number?
Smallest Perfect Number is 6 having divisors 1, 2 and 3.
Can Perfect Number be negative?
No, Perfect Number cannot be negative they are positive integers.
What are the Perfect Number from 1 to 100?
There are only 2 Perfect Number from 1 to 100 which are 6 and 28.
Why is 1 not a Perfect Number?
A Perfect Number can be written as the sum of its divisors (excluding itself) so the only factor of 1 is 1. So, 1 is not a Perfect Number.
How many perfect numbers are there?
In theory there are known to be infinitely many perfect numbers. As of the latest count, 51 perfect numbers have been discovered.
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