Perfect number

2007 Schools Wikipedia Selection. Related subjects: Mathematics

Divisibility-based
sets of integers

Form of factorization:

Prime number

Composite number

Powerful number

Square-free number

Achilles number

Constrained divisor sums:

Perfect number

Almost perfect number

Quasiperfect number

Multiply perfect number

Hyperperfect number

Unitary perfect number

Semiperfect number

Primitive semiperfect number

Practical number

Numbers with many divisors:

Abundant number

Highly abundant number

Superabundant number

Colossally abundant number

Highly composite number

Superior highly composite number

Other:

Deficient number

Weird number

Amicable number

Sociable number

Sublime number

Harmonic divisor number

Frugal number

Equidigital number

Extravagant number

Even perfect numbers

Euclid discovered that the first four perfect numbers are generated by the formula 2ⁿ⁻¹(2ⁿ − 1):

for n = 2: 2¹(2² − 1) = 6

for n = 3: 2²(2³ − 1) = 28

for n = 5: 2⁴(2⁵ − 1) = 496

for n = 7: 2⁶(2⁷ − 1) = 8128

Noticing that 2ⁿ − 1 is a prime number in each instance, Euclid proved that the formula 2ⁿ⁻¹(2ⁿ − 1) gives an even perfect number whenever 2ⁿ − 1 is prime (Euclid, Prop. IX.36).

Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2¹¹ − 1 = 2047 = 23 · 89 is not prime and therefore n = 11 does not yield a perfect number. Two other wrong assumptions were:

The fifth perfect number would have five digits since the first four had 1, 2, 3, and 4 digits respectively.
The perfect numbers would alternately end in 6 or 8.

The fifth perfect number ( $33550336 = 2 12 (2 13 - 1)$ ) has 8 digits, thus refuting the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth (8 589 869 056) also ends in a 6. It is straightforward to show the last digit of any even perfect number must be 6 or 8.

In order for $2 n - 1$ to be prime, it is necessary that $n$ should be prime. Prime numbers of the form 2ⁿ − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers.

Two millennia after Euclid, Euler proved that the formula 2ⁿ⁻¹(2ⁿ − 1) will yield all the even perfect numbers. Thus, every Mersenne prime will yield a distinct even perfect number—there is a concrete one-to-one association between even perfect numbers and Mersenne primes. This result is often referred to as the "Euclid-Euler Theorem". As of December 2006 only 44 Mersenne primes are known, which means there are 44 perfect numbers known, the largest being 2^32,582,656 × (2^32,582,657 − 1) with 19,616,714 digits.

The first 39 even perfect numbers are 2ⁿ⁻¹(2ⁿ − 1) for

n = 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 (sequence A000043 in OEIS)

The other 5 known are for n = 20996011, 24036583, 25964951, 30402457, 32582657. As of 2006 it is not known whether there are others between them.

It is still uncertain 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.

Since any even perfect number has the form 2ⁿ⁻¹(2ⁿ − 1), it is a triangular number, and, like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2ⁿ − 1. Furthermore, any even perfect number except the first one is the sum of the first 2^(n−1)/2 odd cubes:

$6 = 2^1(2^2-1) = 1+2+3, \,$

$28 = 2^2(2^3-1) = 1+2+3+4+5+6+7 = 1^3+3^3, \,$

$496 = 2^4(2^5-1) = 1+2+3+\cdots+29+30+31 = 1^3+3^3+5^3+7^3, \,$

$8128 = 2^6(2^7-1) = 1+2+3+\cdots+125+126+127 = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3. \,$

Odd perfect numbers

It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. Also, it has been conjectured that there are no odd Ore's harmonic numbers. If true, this would imply that there are no odd perfect numbers.

Any odd perfect number N must be of the form 12m + 1 or 36m + 9 and satisfy the following conditions:

N is of the form

$N=q^{\alpha} p_1^{2e_1} \ldots p_k^{2e_k},$

where q, p₁, …, p_k are distinct primes and in modulo 4 arithmetic q ≡ α ≡ 1 (Euler).

N has either q^α > 10²⁰ or $p_j^{2e_j}$ > 10²⁰ for some j (Graeme Laurence Cohen 1987).
The smallest prime factor of N is less than (2k + 8) / 3 (where k is the number of prime factors to an even power, as above) (Grün 1952).
The largest prime factor of N is greater than 10⁸ (Takeshi Goto and Yasuo Ohno, 2006).
The second largest prime factor is greater than 10⁴, and the third largest prime factor is greater than 100 (Iannucci 1999, 2000).
N has at least 75 prime factors; and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors. (Nielsen 2006; Kevin Hare 2005).
N is less than $2^{4^{n}}$ (Nielsen 2003).
N does not have $e 1$ ≡ $e 2$ ...≡ $e k$ ≡ 1 ( modulo 3) (McDaniel 1970).
When $e 1 = e 2 = ... = e k = β$ , k is less than or equal to $16β 2 + 4β + 2$ (Yamada 2005).

If N exists, it must be greater than 10³⁰⁰. A proof is expected for 10⁵⁰⁰ soon. See for more information.

In case of $e 1 = e 2 = ... = e k = β$ in the factorization above, there are no odd perfect numbers when $β$ is equal to 1, 2, 3, 5, 6, 8, 11, 12, 17, 24 or 62 (Steuerwald, McDaniel, Kanold, Hagis, Cohen, Williams).　There are no odd perfect numbers when $β$ is of the form 3k+1, from McDaniel's theorem.

Minor results

Even perfect numbers have a very precise form; odd perfect numbers are rare, if indeed they do exist. 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:

Stuyvaert: Every odd perfect number is the sum of two squares. (1896)
Luca: A Fermat number cannot be a perfect number. (2000)
Makowski: The only even perfect number of the form　 $x 3 + 1$ is 28. (1962)
By dividing the definition through by the perfect number N, the reciprocals of the factors of a perfect number N must add up to 2:
- For 6, we have $1 / 6 + 1 / 3 + 1 / 2 + 1 / 1 = 2$ ;
- For 28, we have $1 / 28 + 1 / 14 + 1 / 7 + 1 / 4 + 1 / 2 + 1 / 1 = 2$ , etc.
The number of divisors of a perfect number (whether even or odd) must be even, since N cannot be a perfect square.
- From these two results it follows that every perfect number is an Ore's harmonic number.
Curtiss (1922) uses a greedy algorithm for Egyptian fractions to prove that a perfect number N must have a number of divisors at least proportional to $lnln N$ . A much stronger singly-logarithmic bound would follow from the nonexistence of odd perfect numbers and the known form of even perfect numbers.

Related concepts

The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.

By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence.

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