Perfect number

In

The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols, ${\displaystyle \sigma _{1}(n)=2n}$ where ${\displaystyle \sigma _{1}}$ is the

This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby ${\displaystyle q(q+1)/2}$ is an even perfect number whenever ${\displaystyle q}$ is a prime of the form ${\displaystyle 2^{p}-1}$ for positive integer ${\displaystyle p}$—what is now called a Mersenne prime. Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form.^[3] This is known as the Euclid–Euler theorem.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.

In about 300 BC Euclid showed that if 2^p − 1 is prime then 2^p−1(2^p − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100.^[4] In modern language, Nicomachus states without proof that every perfect number is of the form ${\displaystyle 2^{n-1}(2^{n}-1)}$ where ${\displaystyle 2^{n}-1}$ is prime.

Euclid proved that ${\displaystyle 2^{p-1}(2^{p}-1)}$ is an even perfect number whenever ${\displaystyle 2^{p}-1}$ is prime (Elements, Prop. IX.36).

For example, the first four perfect numbers are generated by the formula ${\displaystyle 2^{p-1}(2^{p}-1),}$ with p a prime number, as follows: $${\displaystyle {\begin{aligned}p=2&:\quad 2^{1}(2^{2}-1)=2\times 3=6\\p=3&:\quad 2^{2}(2^{3}-1)=4\times 7=28\\p=5&:\quad 2^{4}(2^{5}-1)=16\times 31=496\\p=7&:\quad 2^{6}(2^{7}-1)=64\times 127=8128.\end{aligned}}}$$

Prime numbers of the form ${\displaystyle 2^{p}-1}$ are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For ${\displaystyle 2^{p}-1}$ to be prime, it is necessary that p itself be prime. However, not all numbers of the form ${\displaystyle 2^{p}-1}$ with a prime p are prime; for example, 2¹¹ − 1 = 2047 = 23 × 89 is not a prime number.^[a] In fact, Mersenne primes are very rare: of the approximately 4 million primes p up to 68,874,199, ${\displaystyle 2^{p}-1}$ is prime for only 48 of them.^[14]

While Nicomachus had stated (without proof) that all perfect numbers were of the form ${\displaystyle 2^{n-1}(2^{n}-1)}$ where ${\displaystyle 2^{n}-1}$ 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.^[15] It was not until the 18th century that Leonhard Euler proved that the formula ${\displaystyle 2^{p-1}(2^{p}-1)}$ will yield 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

${\displaystyle 2^{p-1}(2^{p}-1)}$

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, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609 and 57885161 (sequence A000043 in the OEIS).^[14]

Four higher perfect numbers have also been discovered, namely those for which p = 74207281, 77232917, 82589933 and 136279841. Although it is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers for p below 109332539. As of October 2024^[update], 52 Mersenne primes are known,^[16] and therefore 52 even perfect numbers (the largest of which is 2^136279840 × (2^136279841 − 1) with 82,048,640 digits). It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes.

As well as having the form ${\displaystyle 2^{p-1}(2^{p}-1)}$, each even perfect number is the ${\displaystyle (2^{p}-1)}$-th triangular number (and hence equal to the sum of the integers from 1 to ${\displaystyle 2^{p}-1}$) and the ${\displaystyle 2^{p-1}}$-th hexagonal number. Furthermore, each even perfect number except for 6 is the ${\displaystyle {\tfrac {2^{p}+1}{3}}}$-th centered nonagonal number and is equal to the sum of the first ${\displaystyle 2^{\frac {p-1}{2}}}$ odd cubes (odd cubes up to the cube of ${\displaystyle 2^{\frac {p+1}{2}}-1}$):

$${\displaystyle {\begin{alignedat}{3}6&=2^{1}(2^{2}-1)&&=1+2+3,\\[8pt]28&=2^{2}(2^{3}-1)&&=1+2+3+4+5+6+7\\&&&=1^{3}+3^{3}\\[8pt]496&=2^{4}(2^{5}-1)&&=1+2+3+\cdots +29+30+31\\&&&=1^{3}+3^{3}+5^{3}+7^{3}\\[8pt]8128&=2^{6}(2^{7}-1)&&=1+2+3+\cdots +125+126+127\\&&&=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}\\[8pt]33550336&=2^{12}(2^{13}-1)&&=1+2+3+\cdots +8189+8190+8191\\&&&=1^{3}+3^{3}+5^{3}+\cdots +123^{3}+125^{3}+127^{3}\end{alignedat}}}$$

Even perfect numbers (except 6) are of the form $${\displaystyle T_{2^{p}-1}=1+{\frac {(2^{p}-2)\times (2^{p}+1)}{2}}=1+9\times T_{(2^{p}-2)/3}}$$

with each resulting triangular number T₇ = 28, T₃₁ = 496, T₁₂₇ = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with T₂ = 3, T₁₀ = 55, T₄₂ = 903, T₂₇₃₀ = 3727815, ...^[17] 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 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers ${\displaystyle 2^{p-1}(2^{p}-1)}$ with odd prime p and, in fact, with all numbers of the form ${\displaystyle 2^{m-1}(2^{m}-1)}$ for odd integer (not necessarily prime) m.

Owing to their form, ${\displaystyle 2^{p-1}(2^{p}-1),}$ every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example:

$${\displaystyle {\begin{array}{rcl}6_{10}=&2^{2}+2^{1}&=110_{2}\\28_{10}=&2^{4}+2^{3}+2^{2}&=11100_{2}\\496_{10}=&2^{8}+2^{7}+2^{6}+2^{5}+2^{4}&=111110000_{2}\\8128_{10}=&\!\!2^{12}+2^{11}+2^{10}+2^{9}+2^{8}+2^{7}+2^{6}\!\!&=1111111000000_{2}\end{array}}}$$

Thus every even perfect number is a pernicious number.

Every even perfect number is also a practical number (cf. Related concepts).

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,^[18] thus implying that no odd perfect number exists, but Euler himself stated: "Whether ... there are any odd perfect numbers is a most difficult question".^[19] More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist.^[20] All perfect numbers are also harmonic divisor numbers, and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1. Many of the properties proved about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.^[21]

Any odd perfect number N must satisfy the following conditions:

• N > 10¹⁵⁰⁰.^[22]
• N is not divisible by 105.^[23]
• N is of the form N ≡ 1 (mod 12) or N ≡ 117 (mod 468) or N ≡ 81 (mod 324).^[24]
• The largest prime factor of N is greater than 10⁸,^[25] and less than ${\displaystyle {\sqrt[{3}]{3N}}.}$ ^[26]
• The second largest prime factor is greater than 10⁴,^[27] and is less than ${\displaystyle {\sqrt[{5}]{2N}}}$.^[28]
• The third largest prime factor is greater than 100,^[29] and less than ${\displaystyle {\sqrt[{6}]{2N}}.}$^[30]
• N has at least 101 prime factors and at least 10 distinct prime factors.^[22][31] If 3 does not divide N, then N has at least 12 distinct prime factors.^[32]
• N is of the form

${\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},}$

where:

• q, p₁, ..., p_k are distinct odd primes (Euler).
• q ≡ α ≡ 1 (mod 4) (Euler).
• The smallest prime factor of N is at most ${\displaystyle {\frac {k-1}{2}}.}$^[33]
• At least one of the prime powers dividing N exceeds 10⁶².^[22]
• ${\displaystyle N<2^{(4^{k+1}-2^{k+1})}}$^[34][35]
• ${\displaystyle \alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {99k-224}{37}}}$.^[33][36][37]
• ${\displaystyle qp_{1}p_{2}p_{3}\cdots p_{k}<2N^{\frac {17}{26}}}$.^[38]
• ${\displaystyle {\frac {1}{q}}+{\frac {1}{p_{1}}}+{\frac {1}{p_{2}}}+\cdots +{\frac {1}{p_{k}}}<\ln 2}$.^[39][40]

Furthermore, several minor results are known about the exponents e₁, ..., e_k.

• Not all e_i ≡ 1 (mod 3).^[41]
• Not all e_i ≡ 2 (mod 5).^[42]
• If all e_i ≡ 1 (mod 3) or 2 (mod 5), then the smallest prime factor of N must lie between 10⁸ and 10¹⁰⁰⁰.^[42]
• More generally, if all 2e_i+1 have a prime factor in a given finite set S, then the smallest prime factor of N must be smaller than an effectively computable constant depending only on S.^[42]
• If (e₁, ..., e_k) =  (1, ..., 1, 2, ..., 2) with t ones and u twos, then ${\displaystyle (t-1)/4\leq u\leq 2t+{\sqrt {\alpha }}}$.^[43]
• (e₁, ..., e_k) ≠ (1, ..., 1, 3),^[44] (1, ..., 1, 5), (1, ..., 1, 6).^[45]
• If e₁ = ... = e_k = e, then
  • e cannot be 3,^[46] 5, 24,^[47] 6, 8, 11, 14 or 18.^[45]
  • ${\displaystyle k\leq 2e^{2}+8e+2}$ and ${\displaystyle N<2^{4^{2e^{2}+8e+3}}}$.^[48]

In 1888, Sylvester stated:^[49]

... a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number]—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.

On the other hand, several odd integers come close to being perfect. René Descartes observed that the number D = 3² ⋅ 7² ⋅ 11² ⋅ 13² ⋅ 22021 = (3⋅1001)² ⋅ (22⋅1001 − 1) = 198585576189 would be an odd perfect number if only 22021 (= 19² ⋅ 61) 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.

All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:

• The only even perfect number of the form n³ + 1 is 28 (Makowski 1962).^[50]
• 28 is also the only even perfect number that is a sum of two positive cubes of integers (Gallardo 2010).^[51]
• The reciprocals of the divisors of a perfect number N must add up to 2 (to get this, take the definition of a perfect number, ${\displaystyle \sigma _{1}(n)=2n}$, and divide both sides by n):
  • For 6, we have ${\displaystyle {\frac {1}{6}}+{\frac {1}{3}}+{\frac {1}{2}}+{\frac {1}{1}}={\frac {1}{6}}+{\frac {2}{6}}+{\frac {3}{6}}+{\frac {6}{6}}={\frac {1+2+3+6}{6}}={\frac {2\cdot 6}{6}}=2}$;
  • For 28, we have ${\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}$, etc.
• The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.^[52]
  • From these two results it follows that every perfect number is an
    Ore's harmonic number
    .
• The even perfect numbers are not
  trapezoidal numbers; that is, they cannot be represented as the difference of two positive non-consecutive triangular numbers
  . There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form ${\displaystyle 2^{n-1}(2^{n}+1)}$ formed as the product of a
  Fermat prime
  ${\displaystyle 2^{n}+1}$ with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes.[53]
• The number of perfect numbers less than n is less than ${\displaystyle c{\sqrt {n}}}$, where c > 0 is a constant.^[54] In fact it is ${\displaystyle o({\sqrt {n}})}$, using
  little-o notation.^[55]
• Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1 in base 9.[56]^[57] Therefore, in particular the digital root of every even perfect number other than 6 is 1.
• The only square-free perfect number is 6.^[58]

The sum of

associated with a perfect number is a constant sequence. All perfect numbers are also ${\displaystyle {\mathcal {S}}}$-perfect numbers, or

A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers.

• Hyperperfect number
• Leinster group
• List of Mersenne primes and perfect numbers
• Multiply perfect number
• Superperfect numbers
• Unitary perfect number
• Harmonic divisor number