Perfect Numbers in Mathematics

What are Perfect Numbers ?

Perfect numbers are the natural numbers, whose sum of positive divisors (excluding the number itself) is equal to the number itself.Also, it means that sum of all the factors of the number (including itself) is twice the number.

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.
List of first few perfect numbers are

6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 31 + 62 + 124 + 248

Euclid discovered that the first four perfect numbers are generated by the formula 2^p−1(2^p−1), with p a prime number:
for p = 2: 2¹(2²−1) = 6
for p = 3: 2²(2³−1) = 28
for p = 5: 2⁴(2⁵−1) = 496
for p = 7: 2⁶(2⁷−1) = 8128.

Noticing that in each of these cases 2^p−1 is a prime number, Euclid proved that 2^p−1(2^p−1) is an even perfect number whenever 2^p−1 is prime .

Perfect Number Chart

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