Perfect number

A number is called a **perfect number** if by adding all the positive divisors of the number (except itself), the result is the number itself.

Six is the first perfect number. Its divisors (other than the number itself: six) are one, two, and three and one plus two plus three equals six. Other perfect numbers include 28, 496 and 8128.

## Perfect numbers that are even

Euclid discovered that the first four perfect numbers are generated by the formula 2<sup>_n_-1</sup>(2<sup>_n_</sup> - 1):

- for _n_ = 2:   2<sup>1</sup>(2<sup>2</sup> - 1) = 6
- for _n_ = 3:   2<sup>2</sup>(2<sup>3</sup> - 1) = 28
- for _n_ = 5:   2<sup>4</sup>(2<sup>5</sup> - 1) = 496
- for _n_ = 7:   2<sup>6</sup>(2<sup>7</sup> - 1) = 8128

Euclid saw that 2<sup>_n_</sup> - 1 is a prime number in these four cases.  He then proved that the formula 2<sup>_n_-1</sup>(2<sup>_n_</sup> - 1) gives an even perfect number whenever 2<sup>_n_</sup> - 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 two, three, five, and seven are precisely the first four primes, the fifth perfect number would be obtained when _n_ = 11, the fifth prime. However, 2<sup>11</sup> - 1 = 2047 = 23 × 89 is not prime and therefore _n_ = 11 does not give a perfect number. Two other wrong assumptions were:

- The fifth perfect number would have five digits since the first four had one, two, three, and four digits respectively.
- The perfect numbers would alternately end in six or eight.

The fifth perfect number () has eight digits. This falsifies the first assumption. For the second assumption, the fifth perfect number indeed ends with a six. However, the sixth (8,589,869,056) also ends in a six. It is straightforward to show the last digit of any even perfect number must be six or eight.

In order for  to be prime, it is necessary that  should be prime.  Prime numbers of the form 2<sup>_n_</sup> - 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 2<sup>_n_-1</sup>(2<sup>_n_</sup> - 1) will yield all the even perfect numbers. Therefore, 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". Till January 2013, only 48 Mersenne primes are known. This means there are 48 perfect numbers known, the largest being 2<sup>57,885,160</sup> × (2<sup>57,885,161</sup> - 1) with 34,850,340 digits.

The first 42 even perfect numbers are 2<sup>_n_-1</sup>(2<sup>_n_</sup> - 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, 20996011, 24036583, 25964951
The other seven known are for _n_ = 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281.
It is currently not known whether there are others between them.

It is still not known if 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 2<sup>_n_-1</sup>(2<sup>_n_</sup> - 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: 2<sup>_n_</sup> - 1.  Also, any even perfect number except the first one is the sum of the first 2<sup>(n-1)/2</sup> odd cubes:

## Perfect numbers that are odd

It is not known 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.http://oddperfect.org/pomerance.html  Also, it has been conjectured that there are no odd Ore's harmonic numbers. If true, this would mean that there are no odd perfect numbers.

Any odd perfect number _N_ must satisfy the following conditions:
- _N_ > 10<sup>300</sup>.  It is likely that, in the near future, it will be proven that _N_ > 10<sup>500</sup>. http://www.oddperfect.org
- _N_ is of the form
- where:
    - _q_, _p_<sub>1</sub>, ..., _p_<sub>_k_</sub> are distinct primes.
    - _q_ ≡ α ≡ 1 (modulo 4) (Euler).

### Proof

Let  be odd perfect number. Since divisor function is multiplicative,
.

must be an even not divisible by 4 and all the remaining must be odd.

forces .
- Either _q_<sup>α</sup> > 10<sup>20</sup>, or  > 10<sup>20</sup> for some _j_ (Cohen 1987).
- _N_ <  (Nielsen 2003).
- The relation ≡ ...≡  ≡  1 (modulo 3) is not satisfied (McDaniel 1970).
- The smallest prime factor of _N_ is less than (2_k_ + 8) / 3 (Grün 1952).

- The largest prime factor of _N_ is greater than 10<sup>8</sup> (Takeshi Goto and Yasuo Ohno, 2006).
- The second largest prime factor is greater than 10<sup>4</sup>, 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).

## 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:
- Every odd perfect number is of the form 12m + 1 or 4356m + 1089 or 468m + 117 or 2916m + 729 (Roberts 2008).
- An odd perfect number is not divisible by 105 (Kühnel 1949).
- Every odd perfect number is the sum of two squares (Stuyvaert 1896).
- A Fermat number cannot be a perfect number (Luca 2000).
- The only even perfect number of the form　  is 28 (Makowski 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 ;
    - For 28, we have , 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.

## 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 sum-of-divisors function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence.

## References

- Graeme L. Cohen, On the largest component of an odd perfect number, _Journal of the Australian Mathematical Society_, vol. 42 (1987), no. 2, pp. 280–286.
- Euclid, _Elements_, Book IX, Proposition 36. See D.E. Joyce's website for a translation and discussion of this proposition and its proof.
- Takeshi Goto and Yasuo Ohno, _Odd perfect numbers have a prime factor exceeding 10<sup>8</sup>._ Preprint, 2006. Available from Takeshi Goto's webpage "Largest prime factor of an odd perfect number ".
- Otto Grün, Über ungerade vollkommene Zahlen, _Mathematische Zeitschrift_, vol. 55 (1952), pp. 353–354.
- Kevin Hare, _New techniques for bounds on the total number of prime factors of an odd perfect number._  Preprint, 2005. Available from his webpage .
- Douglas E. Iannucci, The second largest prime divisor of an odd perfect number exceeds ten thousand, _Mathematics of Computation_, vol. 68 (1999), no. 228, pp. 1749–1760.
- Douglas E. Iannucci, The third largest prime divisor of an odd perfect number exceeds one hundred, _Mathematics of Computation_, vol. 69 (2000), no. 230, pages 867–879.
- H.-J. Kanold, Untersuchungen über ungerade vollkommene Zahlen, _Journal für die Reine und Angewandte Mathematik_, vol. 183 (1941), pp. 98–109.
- Ullrich Kühnel, Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen, "Mathematische Zeitschrift", vol. 52 (1949), 201—211.
- Florian Luca, The anti-social Fermat number, _Amer. Math. Monthly_ 107 (2000), pp. 171–173.
- W. L. McDaniel, The non-existence of odd perfect numbers of a certain form, _Archiv der Mathematik_ (Basel), vol. 21 (1970), pp. 52–53.
- Pace P. Nielsen, "An upper bound for odd perfect numbers," _Integers_, vol. 3 (2003), A14, 9 pp.
- Pace P. Nielsen, Odd perfect numbers have at least nine different prime factors, _Mathematics of Computation_, in press, 2006.
- T. Roberts, "On the Form of an Odd Perfect Number", _Australian Mathematical Gazette_, 35:4 (2008), p244.
- R. Steuerwald, Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl, _S.-B. Bayer. Akad. Wiss._ 1937, pp. 69–72.
- Tomohiro Yamada, Odd perfect numbers of a special form, _Colloq. Math._ vol. 103 (2005), pp. 303–307.

## Other websites

- David Moews: Perfect, amicable and sociable numbers
- Perfect numbers - History and Theory
- Perfect Number - from MathWorld
- List of Perfect Numbers  at the On-Line Encyclopedia of Integer Sequences
- List of known Perfect Numbers  All known perfect numbers are here.
- OddPerfect.org  A projected distributed computing project to search for odd perfect numbers.