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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.
How perfect numbers related to prime numbers?
They really are not
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.
Are prime numbers more or less common than perfect squares?
Any number squared except 0 is a perfect square so it follows that prime numbers are less common than perfect squares.
Are the perfect and prime numbers same?
No because technically its different .
Do prime numbers have a perfect square a factor?
Just 1.
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.
What is the smallest perfect square that is divisible by the four smallest prime numbers?
44,100
Are the numbers 25 and 49 relativley prime numbers?
No - prime numbers are numbers that can only be divided by 1 and itself. 25 and 49 are examples of perfect squares 5*5 = 25 and 7*7=49
What is a number wit two identical factors?
The prime numbers: 2, 3, 5, 7, 11, 13, 17, 19
