Euclid proved that 2^{p-1}(2^p-1) is an even perfect number whenever 2^p-1 is prime in
Elements (Book IX, Proposition 36). For example, the first four perfect numbers are generated by the formula 2^{p-1}(2^p-1), with a
prime number, as follows: \begin{align} p = 2 &: \quad 2^1(2^2 - 1) = 2 \times 3 = 6 \\ p = 3 &: \quad 2^2(2^3 - 1) = 4 \times 7 = 28 \\ p = 5 &: \quad 2^4(2^5 - 1) = 16 \times 31 = 496 \\ p = 7 &: \quad 2^6(2^7 - 1) = 64 \times 127 = 8128. \end{align} Prime numbers of the form 2^p-1 are known as
Mersenne primes, after the seventeenth-century monk
Marin Mersenne, who studied
number theory and perfect numbers. For 2^p-1 to be prime, it is necessary that itself be prime. However, not all numbers of the form 2^p-1 with a prime are prime; for example, is not a prime number. In fact, Mersenne primes are very rare: of the approximately 4 million primes up to 68,874,199, 2^p-1 is prime for only 48 of them. While
Nicomachus had stated (without proof) that perfect numbers were of the form 2^{n-1}(2^n-1) where 2^n-1 is prime (though he stated this somewhat differently),
Ibn al-Haytham (Alhazen) circa AD 1000 was unwilling to go that far, declaring instead (also without proof) that the formula yielded only every even perfect number. It was not until the 18th century that
Leonhard Euler proved that the formula 2^{p-1}(2^p-1) indeed yields all the even perfect numbers. Thus, there is a
one-to-one correspondence 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. An exhaustive search by the
GIMPS distributed computing project has shown that the first 50 even perfect numbers are 2^{p-1}(2^p-1) for : = 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, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917 . and therefore 52 even perfect numbers (the largest of which is with 82,048,640 digits). It is
not known whether there are
infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes. that even perfect numbers are
triangular As well as having the form 2^{p-1}(2^p-1), each even perfect number is the (2^p-1)-th
triangular number (and hence equal to the sum of the integers from 1 to 2^p-1) and the 2^{p-1}-th
hexagonal number. Furthermore, each even perfect number except for 6 is the \tfrac{2^p+1}{3}-th
centered nonagonal number and is equal to the sum of the first 2^\frac{p-1}{2} odd cubes (odd cubes up to the cube of 2^\frac{p+1}{2}-1): \begin{alignat}{3} 6 &= 2^1(2^2 - 1) &&= 1 + 2 + 3, \\[8pt] 28 &= 2^2(2^3 - 1) &&= 1 + 2 + 3 + 4 + 5 + 6 + 7 \\ & &&= 1^3 + 3^3 \\[8pt] 496 &= 2^4(2^5 - 1) &&= 1 + 2 + 3 + \cdots + 29 + 30 + 31 \\ & &&= 1^3 + 3^3 + 5^3 + 7^3 \\[8pt] 8128 &= 2^6(2^7 - 1) &&= 1 + 2 + 3 + \cdots + 125 + 126 + 127 \\ & &&= 1^3 + 3^3 + 5^3 + 7^3 + 9^3 + 11^3 + 13^3 + 15^3 \\[8pt] 33550336 &= 2^{12}(2^{13} - 1) &&= 1 + 2 + 3 + \cdots + 8189 + 8190 + 8191 \\ & &&= 1^3 + 3^3 + 5^3 + \cdots + 123^3 + 125^3 + 127^3 \end{alignat} Even perfect numbers (except 6) are of the form T_{2^p - 1} = 1 + \frac{(2^p - 2) \times (2^p + 1)}{2} = 1 + 9 \times T_{(2^p - 2)/3} with each resulting triangular number , , (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with , , , It follows that by 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 , , and . This works with all perfect numbers 2^{p-1}(2^p-1) with odd prime and, in fact, with numbers of the form 2^{m-1}(2^m-1) for odd integer (not necessarily prime) . Owing to their form, 2^{p-1}(2^p-1), every even perfect number is represented in binary form as ones followed by zeros; for example: \begin{array}{rcl} 6_{10} =& 2^2 + 2^1 &= 110_2 \\ 28_{10} =& 2^4 + 2^3 + 2^2 &= 11100_2 \\ 496_{10} =& 2^8 + 2^7 + 2^6 + 2^5 + 2^4 &= 111110000_2 \\ 8128_{10} =& \!\! 2^{12} + 2^{11} + 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 \!\! &= 1111111000000_2 \end{array} Thus every even perfect number is a
pernicious number. Every even perfect number is also a
practical number. == Odd perfect numbers == It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496,
Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists, but Euler himself stated: "Whether ... there are any odd perfect numbers is a most difficult question". •
N is not divisible by 105. •
N is of the form
N ≡ 1 (mod 12) or
N ≡ 117 (mod 468) or
N ≡ 81 (mod 324). • The largest
prime power pa that divides
N is greater than 1062. and less than \sqrt[3]{3N}. • The second largest prime factor is greater than 104, and less than \sqrt[5]{2N}. • The third largest prime factor is greater than 100, and less than \sqrt[6]{2N}. •
N has at least 101 prime factors and at least 10 distinct prime factors. If 3 does not divide
N, then
N has at least 12 distinct prime factors. •
N is of the form ::N=q^{\alpha} p_1^{2e_1} \cdots p_k^{2e_k}, :where: :*
q,
p1, ...,
pk are distinct odd primes (Euler). :*
q ≡ α ≡ 1 (
mod 4) (Euler). :* The smallest prime factor of
N is at most \frac{k-1}{2}. :* N :* \alpha + 2e_1 + 2e_2 + 2e_3 + \cdots + 2e_k \geq \frac{99k-224}{37} . :* qp_1p_2p_3 \cdots p_k . :* \frac{1}{q} + \frac{1}{p_1} + \frac{1}{p_2} + \cdots + \frac{1}{p_k} . Furthermore, several minor results are known about the exponents
e1, ...,
ek. • Not all
ei ≡ 1 (
mod 3). • Not all
ei ≡ 2 (
mod 5). • If all
ei ≡ 1 (
mod 3) or 2 (
mod 5), then the smallest prime factor of
N must lie between 108 and 101000. • (
e1, ...,
ek) ≠ (1, ..., 1, 3), (1, ..., 1, 5), (1, ..., 1, 6). • If , then •
e cannot be 3, 5, 24, 6, 8, 11, 14 or 18. In 1888,
Sylvester stated: On the other hand, several odd integers come close to being perfect. René Descartes observed that the number would be an odd perfect number if only were a prime number. The odd numbers with this property (they would be perfect if one of their composite factors were prime) are the
Descartes numbers. Many of the properties proven about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of these numbers may lead to a proof that no odd perfect numbers exist. == Minor results ==