A perfect number is one which equals the sum of its proper divisors. If the sum

Question

A perfect number is one which equals the sum of its proper divisors. If the sum exceeds the number it is abundant, otherwise it is deficient. For example: the number 18 is abundant since

18 is less than 1 + 2 +3 + 6 + 9 =21,

the number 15 is deficient since 15 is greater than 1+3+5 =9 and

6 =1+ 2 + 3 is perfect.

Perfect numbers have been studied since antiquity. It is known that all even perfect numbers are of the form

where both p and

are prime. Primes of that form are called Mersenne Primes.

How many even perfect numbers are there? More precisely, are there infinitely many, or finitely many? In the latter case, just how many are there?

Are there any odd perfect numbers?

Explanation / Answer

1) The answer has not been solved till date. Till 2002 39 Mersenne Primes are known so there are thus 39 known even perfect numbers.

2.) No since odd no has its mulptile1 as perfect no. it ceate some problem.

ex:- 3= 1*3

1+3+= 4 greater than 3

5 = 1*5

1+5 =6 greater than 5

9 = 1*3*3

sum : 3+3+1=7 less than 9

thus it will either greater than or less than .

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