Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding itself. For instance, 28 is a perfect number because the sum of its divisors (1, 2, 4, 7, and 14) is 28.

Perfect numbers are also known as "Complete Numbers" in number theory.
Some of the first perfect numbers are 6, 28, 496, and 8128,
As of 2025, a total of 52 perfect numbers have been discovered.

Other Examples

6
28
496
8128
33550336 up to infinity.

The latest Perfect Number was discovered in 2024 and has 82,048,64 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. 45^th known Mersenne prime, discovered in 2008, is (2^37156667 − 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 respective Perfect Numbers.

Prime, (p)	Mersenne Prime, (2^p -1)	Perfect Number, {2^p-1(2^p -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

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)(2^p − 1)] where 2^p-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:

2^p-1(2^p-1) is an even perfect number where we have 2^p-1 as a prime.

Similarly, we can generate the first four Perfect Numbers using the above formula (p is a 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

List of All 52 Perfect Numbers

Below is a list of all the 52 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
52	388692...008576	82,048,640

These numbers follow the pattern [2^p-1(2^p -1)] where 2^p−1 is a prime number.

Mersenne Prime Numbers

Perfect Number Table

Euclid's Perfect Number Theorem

List of All 52 Perfect Numbers

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