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Perfect number

A perfect number equals the sum of its proper positive divisors. This article explains the definition, key properties, history, examples, and open problems around even and (unknown) odd perfect numbers.

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Overview

A perfect number is an integer equal to the sum of its proper positive divisors — that is, all positive divisors excluding the number itself. In elementary terms, if the divisors of n other than n add up to n, then n is called perfect. The concept belongs to classical number theory and sits alongside related classifications such as deficient and abundant numbers.

Basic properties and examples

Small examples are helpful to illustrate the idea. The first perfect number is 6 because its proper divisors are 1, 2 and 3 and 1 + 2 + 3 = 6. Other well-known small perfect numbers include 28, 496 and 8128. Perfect numbers are rare compared with ordinary integers and exhibit several characteristic algebraic features.

Mathematical characterization

Even perfect numbers are completely described by a classical result: whenever 2^p − 1 is a prime (a Mersenne prime), the number 2^{p-1}(2^p − 1) is perfect. Conversely, every even perfect number has this form. Thus there is a close relationship between perfect numbers and Mersenne primes. The proper divisors that sum to the perfect number include powers of two and the Mersenne prime factor; in the divisor-sum computation each divisor other than the number itself contributes to the total.

History and computation

The study of perfect numbers dates back to antiquity and was considered by Greek mathematicians. Later developments tied perfect numbers to the search for Mersenne primes, a focus of both theoretical research and large-scale distributed computation. Modern prime-finding efforts have produced very large Mersenne primes, and each such prime immediately yields a correspondingly large even perfect number.

Open problems and distinctions

No odd perfect number is known. Despite extensive work, no example has been found and many restrictive necessary conditions have been proved for any hypothetical odd perfect number; these make its existence an open and long-standing problem in number theory. For context, numbers are often classified by comparing the sum of proper divisors with the number itself: if the sum is smaller the number is deficient, if larger it is abundant, and if equal it is perfect. The study of divisors (divisor) functions and their sums remains an active area in multiplicative number theory.

Significance and examples of use

  • Perfect numbers are mainly of theoretical and historical interest within mathematics.
  • Their relation to Mersenne primes links them to computational prime searches and algorithms used in primality testing.
  • They provide simple, illustrative examples when teaching divisor functions and classification of integers.

Classic problems

  • It is open whether there are infinitely many perfect numbers.
  • It is open whether there are infinitely many even perfect numbers. This question coincides with the question whether there are infinitely many Mersenne primes.
  • It is open whether an odd perfect number exists at all. If such a number exists, it has the following properties:
    • It's bigger than 101500.
    • It is of the form 12k+1 or 36k+9with a natural number k. (Theorem of Jacques Touchard).
    • It has at least 8 different prime divisors.
    • It has at least 11 distinct prime divisors if it is not divisible by 3.
    • If is Athe number of its distinct prime divisors and is the p_{1}smallest of them, then p_{1}<{\frac {2A}{3}}+2(Otto Green's theorem).
    • It is less than 4^{{4^{A}}}(Theorem of D. R. Heath-Brown).
    • If it is less than 109118, then it is p^6divisible by with a prime pwhich is greater than 10500.
    • It is not a square number.

Other properties of the perfect numbers

Sum of the reciprocal divisors

The sum of the reciprocals of all divisors of a perfect number n(including the number itself) is 2:

\sum_{k\mid n} \frac{1}{k} = \sum_{k\mid n} \frac{k}{n} = \frac{1}{n}\sigma(n) = \frac{2n}{n} = 2

Example:

For n = 6, {\displaystyle {\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{6}}={\frac {12}{6}}=2}

Illustration by Eaton (1995, 1996)

Every even perfect number n > 6 has the representation

n = 1 + \frac{9}{2}k(k+1)with k = 8j+2and a nonnegative integer j.

Conversely, one does not obtain a perfect number for every natural number . j

Examples:

{\displaystyle j=0}gives k=2and {\displaystyle n=28}{perfect}.

j = 1gives k=10and {\displaystyle n=496}{{perfect}}

j = 2gives {\displaystyle k=18}and {\displaystyle n=1540}(not perfect).

Sum of the cubes of the first odd natural numbers

With the exception of 6, any even perfect number nrepresented as

{\displaystyle n=\sum _{k=1}^{2^{\frac {p-1}{2}}}~(2k-1)^{3},}

where is pthe exponent of the Mersenne prime from the representation . {\displaystyle n=2^{p-1}(2^{p}-1)}

Examples:

28 = 1^3 + 3^3

496 = 1^3 + 3^3 + 5^3 + 7^3

Remark:
For any {\displaystyle m\in \mathbb {N} }and {\displaystyle q=m^{2}}holds:

{\displaystyle \sum _{k=1}^{m}(2k-1)^{3}=2m^{4}-m^{2}=q\cdot (2q-1)}(summation formula of odd cubes).

In particular, this is also true for all powers of two {\displaystyle m=2^{r}}and {\displaystyle q=(2^{2})^{r}}with {\displaystyle r\in \mathbb {N} }to:

{\displaystyle \sum _{k=1}^{2^{r}}(2k-1)^{3}=2\,(2^{r})^{4}-(2^{r})^{2}=(2^{2})^{r}\,(2\cdot (2^{2})^{r}-1)}

With odd one pcan {\displaystyle r={\frac {p-1}{2}}}substitute

{\displaystyle \sum _{k=1}^{2^{\frac {p-1}{2}}}(2k-1)^{3}=2\,(2^{\frac {p-1}{2}})^{4}-(2^{\frac {p-1}{2}})^{2}=(2^{2})^{\frac {p-1}{2}}\,(2\cdot (2^{2})^{\frac {p-1}{2}}-1)=2^{p-1}\,(2\cdot 2^{p-1}-1)}

{\displaystyle \sum _{k=1}^{2^{\frac {p-1}{2}}}(2k-1)^{3}=2^{p-1}\,(2^{p}-1)}

The representation as a sum of cubic numbers is a property, which is only indirectly related to perfect numbers.

{\displaystyle n=2^{p-1}(2^{p}-1)=6,28,496,8128,33550336,\dotsc }with p = 2, 3, 4, 5, 6, ...

(only after removing the first perfect number n(p=2)=6 and assuming that there are no odd perfect numbers), but a property of the number series

{\displaystyle n^{*}=2m^{4}-m^{2}=0,1,28,153,496,1225,2556,4753,8128,13041,\dotsc }

is. We also see why it cannot hold for the first perfect number ( p=2is not odd and therefore is rnot an integer).
Incidentally, this equation is satisfied
for numbers {\displaystyle n^{*}<10^{50}} in addition to eight perfect numbers out of a total of 2,659,147,948,473 numbers.

Sum of the first natural numbers

Any even perfect number n can be krepresented by an appropriate natural number as

n = \sum_{i=1}^ki = \frac{k(k+1)}{2}

or in other words: Every even perfect number is also a triangular number. As mentioned above, is kalways a Mersenne prime.

Examples:

6 = 1 + 2 + 3 = \frac{3\cdot 4}{2}

28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 = \frac{7\cdot 8}{2}

{\displaystyle 496=1+2+3+4+5+\dotsb +31={\frac {31\cdot 32}{2}}}

{\displaystyle 8128=1+2+3+4+5+\dotsb +127={\frac {127\cdot 128}{2}}}

Another representation

Any even perfect number n can be krepresented by an appropriate natural number as

{\displaystyle n={\binom {2^{k}}{2}}.}

Binary System

An even perfect number appears in the dual system as a characteristic sequence of ones and zeros.

Because of its form {\displaystyle \left(2^{p+1}-1\right)\cdot 2^{p}}, it represents itself in the base-2 number system as a sequence of p+1ones and pzeros:

{\displaystyle 6=110_{2}}

{\displaystyle 28=11100_{2}}

{\displaystyle 496=111110000_{2}}

{\displaystyle 8128=1111111000000_{2}}

{\displaystyle 33550336=1111111111111000000000000_{2}}

Quaternary System

An even perfect number n>6 appears in the quaternary system as a characteristic sequence of threes and zeros.

Given its form {\displaystyle \left(2^{2r+1}-1\right)\cdot 2^{2r}}it represents itself in the base-4 number system as a sequence of 1ones, rthrees, and rzeros:

{\displaystyle 28=130_{4}}

{\displaystyle 496=13300_{4}}

{\displaystyle 8128=1333000_{4}}

{\displaystyle 33550336=1333333000000_{4}}

Questions and answers

Q: What is a perfect number?

A: A perfect number is a number for which the addition of all its positive divisors (excluding itself) equals the number itself.

Q: What is the first perfect number?

A: The first perfect number is 6.

Q: How do you find the divisors of a number?

A: You can find the divisors of a number by dividing the number by integers starting from 1 up to the number itself and counting those that divide evenly with no remainder.

Q: What are the divisors of 6?

A: The divisors of 6 are 1, 2, and 3.

Q: How do you know that 6 is a perfect number?

A: We know that 6 is a perfect number because the sum of its divisors (1 + 2 + 3) equals 6.

Q: What are some examples of other perfect numbers?

A: Other examples of perfect numbers include 28, 496 and 8128.

Q: How many perfect numbers are there?

A: It is not known whether there are an infinite number of perfect numbers, but only 51 are known to exist as of 2021.

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