lpetrich
Contributor
For positive integers n, one can construct a divisor-sum function σ:
\( n = \prod_i (p_i)^{m_i} ;\ \sigma(n) = \prod_i \frac{(p_i)^{m_i} - 1}{p_i - 1} \)
More generally, σk is the sum of powers k of divisors, with σ0 being the number of divisors.
\( \sigma_k(n) = \prod_i \frac{(p_i)^{k m_i} - 1}{(p_i)^k - 1} ;\ \sigma_0(n) = \prod_i m_i \)
A Perfect number is a number n which is the sum of all its proper divisors. That is, σ = 2n, since the sum in σ includes n itself. One may define a restricted divisor sum s = σ - n that omits n. Thus for a perfect number, s = n.
All even perfect numbers are known. They have form 2n-1 * (2n - 1) where (2n - 1) is a prime number, a Mersenne prime In fact, every Mersenne prime has an associated perfect number. For (2n - 1) to be a prime number, n must also be a prime number, though only some prime numbers give Mersenne primes. The smallest one that doesn't is 11, and 2^11-1 = 8191 = 23*89. The previous four, for 2, 3, 5, 7, were known in antiquity: 3, 7, 31, 127, along with their perfect numbers: 6, 28, 496, 8128.
If you want to help search for large Mersenne primes, consider participating in the Great Internet Mersenne Prime Search
It is not known whether or not odd perfect numbers exist, and some strong constraints have been placed on them. They must be greater than 101500, for instance.
\( n = \prod_i (p_i)^{m_i} ;\ \sigma(n) = \prod_i \frac{(p_i)^{m_i} - 1}{p_i - 1} \)
More generally, σk is the sum of powers k of divisors, with σ0 being the number of divisors.
\( \sigma_k(n) = \prod_i \frac{(p_i)^{k m_i} - 1}{(p_i)^k - 1} ;\ \sigma_0(n) = \prod_i m_i \)
A Perfect number is a number n which is the sum of all its proper divisors. That is, σ = 2n, since the sum in σ includes n itself. One may define a restricted divisor sum s = σ - n that omits n. Thus for a perfect number, s = n.
All even perfect numbers are known. They have form 2n-1 * (2n - 1) where (2n - 1) is a prime number, a Mersenne prime In fact, every Mersenne prime has an associated perfect number. For (2n - 1) to be a prime number, n must also be a prime number, though only some prime numbers give Mersenne primes. The smallest one that doesn't is 11, and 2^11-1 = 8191 = 23*89. The previous four, for 2, 3, 5, 7, were known in antiquity: 3, 7, 31, 127, along with their perfect numbers: 6, 28, 496, 8128.
If you want to help search for large Mersenne primes, consider participating in the Great Internet Mersenne Prime Search
It is not known whether or not odd perfect numbers exist, and some strong constraints have been placed on them. They must be greater than 101500, for instance.