Perfect number

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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.

The first perfect number is 6, 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 (sequence A000396 in OEIS).

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


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[edit] 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 = 212(213 − 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 2n − 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 232,582,656 × (232,582,657 − 1) with 19,616,714 digits. 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. \,

[edit] 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. [1] 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, …, pk are distinct primes and in modulo 4 arithmatic q ≡ α ≡ 1 (Euler).
  • N has either qα > 1020 or p_j^{2e_j} > 1020 for some j (Graeme Laurence Cohen 1987).
  • The smallest prime factor of N is less than (2n + 6) / 3 where n is the number of distinct prime factors (so with k as used above, n = k + 1) (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 e1≡e2...≡ek ≡ 1 (modulo 3) (McDaniel 1970).

If N exists, it must be greater than 10500. See [2] for more information.

In case of e1 = e2 = ... = ek = β in the factorization above, Tomohiro Yamada proved that there are only finitely many odd perfect numbers for any given β. 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).[citation needed]　There are no odd perfect numbers when β is of the form 3k+1, from McDaniel's theorem.

[edit] 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　 x3 + 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.
  • 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 lnlnN. A much stronger singly-logarithmic bound would follow from the nonexistence of odd perfect numbers and the known form of even perfect numbers.

[edit] 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.

[edit] See also

[edit] 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.
  • D. R. Curtiss, On Kellogg's diophantine problem (via JSTOR), American Mathematical Monthly, vol. 29 (1922), pp. 380–387.
  • 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. 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.
  • W. L. McDaniel, The non-existence of odd perfect numbers of a certain form, Archiv der Mathematik (Basel), vol. 21 (1970), 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. arXiv:math.NT/0602485.
  • R. Steuerwald, Verscharfung einen notwendigen Bedingung fur die Existenz einen ungeraden vollkommenen Zahl, S.-B. Bayer. Akad. Wiss. 1937, 69–72.
  • Tomohiro Yamada, Odd perfect numbers of a special form, Colloq. Math. vol. 103 (2005), 303–307.

[edit] External links

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