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Perfect numbers cannot be prime numbers. Here's why:
A number N is perfect if σ(N) = 2N (σ is the sum of divisors function). If there is a prime p that is a perfect number, then σ(p) = 2p. However, the only factors of p are 1 and p, so σ(p) is also equal to p+1. If 2p = p+1, then p=1, which is not prime, and 1 is defined to have only one factor, 1.
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Why are no prime numbers perfect numbers?
Prime numbers have two factors. The sum of their proper divisors is always 1.
What is the relationship between Mersenne prime and perfect numbers?
Perfect numbers have the form 2n-1(2n-1) where 2n-1 is a Mersenne prime. When a new Mersenne prime is discovered, so is a new perfect number.
Is it possible for a prime number to be a perfect number and why?
No, all prime numbers are deficient.
What square numbers below 100 are also prime numbers?
A [perfect] square number, by definition, has a factor which is its square root. As a result it CANNOT be a prime!
How many perfect numbers are known?
So far 47. Euler proved that every even perfect number will be of the form 2p−1(2p−1), where p is prime and 2p−1 is also prime. If 2p−1 is prime it is known as a Mersenne prime. Since 47 Mersenne primes are known, 47 even perfect numbers are known. As for odd perfect numbers, none are known, nor has it been proven yet that there aren't any.
