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Perfect number

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Divisibility-based
   sets of integers
   Form of factorization:
   Prime number
   Composite number
   Powerful number
   Square-free number
   Achilles number
   Constrained divisor sums:
   Perfect number
   Almost perfect number
   Quasiperfect number
   Multiply perfect number
   Hyperperfect number
   Unitary perfect number
   Semiperfect number
   Primitive semiperfect number
   Practical number
   Numbers with many divisors:
   Abundant number
   Highly abundant number
   Superabundant number
   Colossally abundant number
   Highly composite number
   Superior highly composite number
   Other:
   Deficient number
   Weird number
   Amicable number
   Sociable number
   Sublime number
   Harmonic divisor number
   Frugal number
   Equidigital number
   Extravagant number
   See also:
   Divisor function
   Divisor
   Prime factor
   Factorization

   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.

Even perfect numbers

   Euclid discovered that the first four perfect numbers are generated by
   the formula 2^n−1(2^n − 1):

          for n = 2:   2^1(2^2 − 1) = 6
          for n = 3:   2^2(2^3 − 1) = 28
          for n = 5:   2^4(2^5 − 1) = 496
          for n = 7:   2^6(2^7 − 1) = 8128

   Noticing that 2^n − 1 is a prime number in each instance, Euclid proved
   that the formula 2^n−1(2^n − 1) gives an even perfect number whenever
   2^n − 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, 2^11 − 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
   refuting 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 2^n − 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^n−1(2^n − 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
   2^32,582,656 × (2^32,582,657 − 1) with 19,616,714 digits.

   The first 39 even perfect numbers are 2^n−1(2^n − 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
          (sequence A000043 in OEIS)

   The other 5 known are for n = 20996011, 24036583, 25964951, 30402457,
   32582657. As of 2006 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 2^n−1(2^n − 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^n − 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 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. 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, p[1], …, p[k] are distinct primes and in modulo 4
          arithmetic q ≡ α ≡ 1 (Euler).

     * N has either q^α > 10^20 or p_j^{2e_j} > 10^20 for some j (Graeme
       Laurence Cohen 1987).
     * The smallest prime factor of N is less than (2k + 8) / 3 (where k
       is the number of prime factors to an even power, as above) (Grün
       1952).
     * The largest prime factor of N is greater than 10^8 (Takeshi Goto
       and Yasuo Ohno, 2006).
     * The second largest prime factor is greater than 10^4, 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 e[1]≡e[2]...≡e[k] ≡ 1 ( modulo 3) (McDaniel 1970).
     * When e[1] = e[2] = ... = e[k] = β, k is less than or equal to 16β^2
       + 4β + 2 (Yamada 2005).

   If N exists, it must be greater than 10^300. A proof is expected for
   10^500 soon. See for more information.

   In case of e[1] = e[2] = ... = e[k] = β in the factorization above,
   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).　There are no odd perfect numbers when β is of the form 3k+1,
   from McDaniel's theorem.

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)
     * Luca: A Fermat number cannot be a perfect number. (2000)
     * 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.
     * 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.

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.

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