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 σ_{1}(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.
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Even perfect numbers
Euclid discovered that the first four perfect numbers are generated by the formula 2^{p−1}(2^{p}−1), with p a prime number:
Noticing that in each of these cases 2^{p}−1 is a prime number, Euclid proved that 2^{p−1}(2^{p}−1) is an even perfect number whenever 2^{p}−1 is prime (Euclid, Prop. IX.36).
In order for 2^{p}−1 to be prime, it is necessary that p itself be prime. Prime numbers of the form 2^{p}−1 are known as Mersenne primes, after the seventeenthcentury monk Marin Mersenne, who studied number theory and perfect numbers. However, not all numbers of the form 2^{p}−1 with p a prime are prime; for example, 2^{11}−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, 2^{p}−1 is prime for only 33 of them.
Over a millennium after Euclid, Ibn alHaytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of the form 2^{p−1}(2^{p}−1) where 2^{p}−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 2^{p−1}(2^{p}−1) will yield all the even perfect numbers. Thus, there is a onetoone 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^{[update]}, 47 Mersenne primes and therefore 47 even perfect numbers are known.^{[2]} The largest of these is 2^{43,112,608} × (2^{43,112,609}−1) with 25,956,377 digits.
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