Perfect Number - Even Perfect Numbers

Even Perfect Numbers

Euclid proved that 2p−1(2p−1) is an even perfect number whenever 2p−1 is prime (Euclid, Prop. IX.36).

For example, the first four perfect numbers are generated by the formula 2p−1(2p−1), with p a prime number, as follows:

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.

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 a prime p are prime; for example, 211−1 = 2047 = 23 × 89 is not a prime number. (All factors of 2p−1 will be congruent to 1 mod 2p. For example, 211−1 = 2047 = 23 × 89 and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, whenever p is a Sophie Germain prime—that is, 2p+1 is also prime—and 2p+1 is congruent to 1 or 7 mod 8, then 2p + 1 will be a factor of 2p−1.) In fact, Mersenne primes are very rare—of the 1,509,263 prime numbers p up to 24,036,583, 2p−1 is prime for only 41 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. 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. The largest of these is 243,112,608 × (243,112,609−1) with 25,956,377 digits.

The first 41 even perfect numbers are 2p−1(2p−1) for

p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, and 24036583 (sequence A000043 in OEIS).

The other 6 known are for p = 25964951, 30402457, 32582657, 37156667, 42643801, and 43112609. It has not yet been proved that there are (or are not) others after the 41st.

No proof is known whether there are infinitely many Mersenne primes and perfect numbers.

The search for new Mersenne primes is the goal of the GIMPS distributed computing project.

Because any even perfect number has the form 2p−1(2p−1), it is the (2p−1)th triangular number and the 2p−1th hexagonal number. Like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2p−1. Furthermore, any even perfect number except the first one is the ((2p+1)/3)th centered nonagonal number as well as the sum of the first 2(p−1)/2 odd cubes:

Even perfect numbers (except 6) give remainder 1 when divided by 9. (In fact, subtracting 1 and dividing the result by 9 always gives a triangular number, the sequence starting with 3, 55, 903, 3727815, ....) This can be reformulated as follows: adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers 2p−1(2p−1) with odd prime p and, in fact, all numbers of the form 2m−1(2m−1) for odd integer (not necessarily prime) m.

Owing to their form, 2p−1(2p−1), every even perfect number is represented in binary as p ones followed by p − 1 zeros:

6₁₀ = 110₂

28₁₀ = 11100₂

496₁₀ = 111110000₂

8128₁₀ = 1111111000000₂

33550336₁₀ = 1111111111111000000000000₂.

Thus every even perfect number is a pernicious number.

Note that every even perfect number is also a practical number (c.f. Related concepts).