Perfect number

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In mathematics, a perfect number is a positive integer that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself), or σ1(n) = 2n.

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.

The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 (sequence A000396 in OEIS).

These first four perfect numbers were the only ones known to early Greek mathematics.


Even perfect numbers

Euclid discovered that the first four perfect numbers are generated by the formula 2p−1(2p−1), with p a prime number:

Noticing that in each of these cases 2p−1 is a prime number, Euclid proved that 2p−1(2p−1) is an even perfect number whenever 2p−1 is prime (Euclid, Prop. IX.36).

In order for 2p−1 to be prime, it is necessary that p itself be 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; for example, 211−1 = 2047 = 23 × 89 is not a prime number. In fact, Mersenne primes are very rare — of the 78,498 prime numbers p below 1,000,000, 2p−1 is prime for only 33 of them.

Over a millennium after Euclid, Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of the form 2p−1(2p−1) where 2p−1 is prime, but he was not able to prove this result.[1] It was not until the 18th century that Leonhard Euler proved that the formula 2p−1(2p−1) will yield all the even perfect numbers. Thus, there is a one-to-one relationship between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler Theorem. As of June 2010, 47 Mersenne primes and therefore 47 even perfect numbers are known.[2] The largest of these is 243,112,608 × (243,112,609−1) with 25,956,377 digits.

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