Perfect number

In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n.

Six (6) is the first perfect number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 .

These first four perfect numbers were the only ones known to the ancient Greeks.

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.

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 debunking 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". Only 43 Mersenne primes are presently known, which means there are 43 perfect numbers known. 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 have 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 q ≡ α ≡ 1 (mod 4) (Euler).

In the above factorization, k is at least 8; and k is at least 11 if 3 does not divide N (Nielsen 2006).
In the above factorization, at least one of the $e_1, e_2, \ldots e_k$ is larger than 1. (Steuerwald 1937)
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 in total, counting each of the 2e_k repetitions of p_k separately (Kevin Hare 2005).
N is less than $2^{4^{n}}$ where n is the number of distinct prime factors (so n = k + 1 with k as above) (Nielsen 2003).

If N exists, it must be greater than 10⁵⁰⁰. See ^* for more information.

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)
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.

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.

References

Takeshi Goto and Yasuo Ohno, Odd perfect numbers have a prime factor exceeding 10⁸. Preprint, 2006. Available from Takeshi Goto's webpage "Largest prime factor of an odd perfect number".
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, volume 68, issue 228, pages 1749–1760, 1999.
Douglas E. Iannucci, "The third largest prime divisor of an odd perfect number exceeds one hundred," Mathematics of Computation, volume 69, issue 230, pages 867–879, 2000.
Pace P. Nielsen, "An upper bound for odd perfect numbers," Integers, vol. 3, A14, 9 pp. (electronic), 2003.
Pace P. Nielsen, Odd perfect numbers have at least nine different prime factors, , 2006.
R. Steuerwald, Verscharfung einen notwendigen Bedingung fur die Existenz einen ungeraden vollkommenen Zahl, S.-B. Bayer. Akad. Wiss. 1937, 69–72.

External links

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.

Integer sequences | Unsolved problems in mathematics