Perfect number

In mathematics, a perfect number is a positive integer that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself), or σ(n) = 2n.

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.

The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 (sequence A000396 in OEIS).

These first four perfect numbers were the only ones known to early Greek mathematics.

Contents

Even perfect numbers

Euclid discovered that the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number:

for p = 2:   21(22 − 1) = 6
for p = 3:   22(23 − 1) = 28
for p = 5:   24(25 − 1) = 496
for p = 7:   26(27 − 1) = 8128.

Noticing that in each of these cases 2p − 1 is a prime number, Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Euclid, Prop. IX.36).

In order for 2p − 1 to be prime, it is necessary that p itself be prime. Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. However, not all numbers of the form 2p − 1 with p a prime are prime.[1] In fact, Mersenne primes are very rare — of the 78,498 prime numbers p below 1,000,000, 2p − 1 is prime for only 33 of them.

Over a millennium after Euclid, Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of the form 2p−1(2p − 1) where 2p − 1 is prime, but he was not able to prove this result.[2] It was not until the 18th century that Leonhard Euler proved that the formula 2p−1(2p − 1) will yield all the even perfect numbers. Thus, there is a one-to-one relationship 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. As of June 2010, 47 Mersenne primes and therefore 47 even perfect numbers are known.[3] The largest of these is 243,112,608 × (243,112,609 − 1) with 25,956,377 digits.

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

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 (sequence A000043 in OEIS).

The other 8 known are for p = 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609. 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 2p−1(2p − 1), it is the (2p − 1)th triangular number and the 2p−1th hexagonal number. Like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2p − 1. Furthermore, any even perfect number except the first one is the sum of the first 2(p−1)/2 odd cubes:

\begin{align} 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. \end{align}

Even perfect numbers (except 6) give remainder 1 when divided by 9. This can be reformulated as follows. 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 is obtained — the resulting number is called the digital root — produces the number 1. For example, the digital root of 8128 = 1, since 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. The reason that this digital root doesn't work with the perfect number 6 is because it only works with perfect numbers 2p−1(2p − 1), with odd prime p; the perfect number 6 being associated with the even prime 2.

Owing to their form, 2p−1(2p − 1), every even perfect number is represented in binary as p ones followed by p − 1  zeros:

610 = 1102
2810 = 111002
49610 = 1111100002
812810 = 11111110000002

Odd perfect numbers

Unsolved problems in mathematics
Are there any odd perfect numbers? Question mark2.svg

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.[4] Also, it has been conjectured that there are no odd Ore's harmonic numbers, except for 1. If true, this would imply that there are no odd perfect numbers.

Any odd perfect number N must satisfy the following conditions:

N=q^{\alpha} p_1^{2e_1} \cdots p_k^{2e_k},
where:
  • q, p1, ..., pk are distinct primes (Euler).
  • q ≡ α ≡ 1 (mod 4) (Euler).
  • The smallest prime factor of N is less than (2k + 8) / 3 (Grün 1952).
  • Either qα > 1020, or p j2ej  > 1020 for some j (Cohen 1987).
  • N < 24k+1 (Nielsen 2003).

In 1888, Sylvester stated:[6]

...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.

Minor results

All 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:

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.

See also

Notes

  1. ↑ E.g. 2^11 − 1 = 2047 = 23 × 89 is not a prime number.
  2. ↑ O'Connor, John J.; Robertson, Edmund F., "Abu Ali al-Hasan ibn al-Haytham", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Haytham.html .
  3. ↑ http://www.mersenne.org/
  4. ↑ Oddperfect.org
  5. ↑ Oddperfect.org
  6. ↑ The Collected Mathematical Papers of James Joseph Sylvester p. 590, tr. from "Sur les nombres dits de Hamilton", Compte Rendu de l'Assoiation Française (Toulouse, 1887), pp. 164–168.
  7. ↑ Jones, Chris; Lord, Nick (1999), "Characterising non-trapezoidal numbers", The Mathematical Gazette (The Mathematical Association) 83 (497): 262–263, doi:10.2307/3619053, http://www.jstor.org/stable/3619053 

References

  • Graeme L. Cohen, "On the largest component of an odd perfect number", Journal of the Australian Mathematical Society, 42:2 (1987), 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 108", Mathematics of Computation 77:263 (2008), pp. 1859–1868. doi:10.1090/S0025-5718-08-02050-9.
  • Otto Grün, "Über ungerade vollkommene Zahlen", Mathematische Zeitschrift, 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, 68:228 (1999), pp. 1749–1760.
  • Douglas E. Iannucci, "The third largest prime divisor of an odd perfect number exceeds one hundred", Mathematics of Computation, 69:230 (2000), pp. 867–879.
  • H.-J. Kanold, "Untersuchungen über ungerade vollkommene Zahlen", Journal für die Reine und Angewandte Mathematik, 183 (1941), pp. 98–109.
  • Ullrich Kühnel, "Verschärfung der notwendigen Bedingungen für die Existenz von ungeraden vollkommenen Zahlen", Mathematische Zeitschrift, 52 (1949), pp. 201–211.
  • Pace P. Nielsen, "An upper bound for odd perfect numbers," Integers, 3 (2003), A14, 9 pp.
  • Pace P. Nielsen, "Odd perfect numbers have at least nine different prime factors", Mathematics of Computation, in press, 2006. arXiv:math.NT/0602485.
  • T. Roberts, "On the Form of an Odd Perfect Number", Australian Mathematical Gazette, 35:4 (2008), p. 244.
  • R. Steuerwald, "Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl", S.-B. Bayer. Akad. Wiss., 1937, pp. 69–72.

Further reading

External links