Definition:
A perfect number is the positive integer in that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently of a perfect number is that a number is half the sum of all of its positive divisors (including itself), or σ(n) = 2n.
Even perfect numbers
In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. However, not all numbers of the form 2p − 1 with p a prime are prime
In the first four perfect numbers are generated by the formula 2p−1(2p − 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 2p − 1 is a prime number in each instance, Euclid proved that the formula 2p−1(2p − 1) gives an even perfect number In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes. I have recently faced lot of problem while learning Definition of Rational Numbers, But thank to online resources of math which helped me to learn myself easily on net.
Odd perfect numbers
It is unknown whether there are any odd perfect numbers. Various of results have been obtained, but none of that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has been presented by a heuristic argument in which suggests that no odd perfect numbers exist.[4] Also, it can been conjectured of that there are no odd Ore's harmonic numbers, except for 1. If true in this would to imply that there are no odd perfect numbers.
A perfect number is the positive integer in that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently of a perfect number is that a number is half the sum of all of its positive divisors (including itself), or σ(n) = 2n.
Even perfect numbers
In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. However, not all numbers of the form 2p − 1 with p a prime are prime
In the first four perfect numbers are generated by the formula 2p−1(2p − 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 2p − 1 is a prime number in each instance, Euclid proved that the formula 2p−1(2p − 1) gives an even perfect number In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes. I have recently faced lot of problem while learning Definition of Rational Numbers, But thank to online resources of math which helped me to learn myself easily on net.
Odd perfect numbers
It is unknown whether there are any odd perfect numbers. Various of results have been obtained, but none of that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has been presented by a heuristic argument in which suggests that no odd perfect numbers exist.[4] Also, it can been conjectured of that there are no odd Ore's harmonic numbers, except for 1. If true in this would to imply that there are no odd perfect numbers.
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