Perfect Number

A number which is equal to the sum of all of its proper divisors is called Perfect number.

Proper divisor:- The positive divisor of a number excluding itself.

Let N be a number and d₁,d₂,d₃.....d_m be the proper divisors of N.

N is a perfect number if

N = d₁ + d₂ + d₃ + ......... + d_m

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Example:  6, 28, 496, 8128, 33550336

The proper divisors of 6 are 1,2 and 3

6=1+2+3

Since 6 is the sum of all of its Proper divisors hence it’s a Perfect Number.

Question: Is 24 a Perfect Number?

The proper divisors of 24 are 1,2,3,4,6,8,12.

24 ≠ 1+2+3+4+6+8+12

Since the sum of proper divisors of 24 are nor equal to 24, It’s not a Perfect Number

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Euclid proved that all even Perfect Number is of the form

N = 2^P(2^P-1)

Where P is a Prime for which (2^P-1) is a Mersenne Prime.

Mersenne Prime:- The Prime number which is of the form 2ⁿ-1; where n is an Integer.

Facts:

• 6 is the smallest Perfect number.
• There are 51 known perfect number.
• The latest Perfect Number is discovered in 2018 which has 49,724,095 digits.

                                                    

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