Diapositiva 1

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Perfect numbers Números perfectos
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Perfect numbers
s(n) n s(6) 1 2 3 6.
Los pitagóricos fueron los primeros en
preocuparse por ellos. San Agustín (354-430) en
"La ciudad de Dios" dice que Dios prefirió crear
el mundo en 6 días porque 6 significa perfección
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The smallest perfect numbers are
6 known to the Greeks 28 known to the
Greeks 496 known to the Greeks 8.128 known to
the Greeks 33.550.336 recorded in medieval
manuscript 8.589.869.056 Cataldi found in
1588 137.438.691.328 Cataldi found in 1588
Sequence A000396 in OEIS
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Some Interesting Facts about Perfect Numbers
Euclid discovered that the first four perfect
numbers are generated by the formula
2p-1(2p - 1) 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
2p - 1 is a prime number in each instance. Euclid
proved that the formula 2p-1(2p - 1) gives a
perfect number whenever 2p - 1 is prime. (Euclid,
Prop. IX.36).
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• EuclidEuler Theorem
• Euler proved that any even perfect number is
  given by the formula 2p-1 (2p-1), where 2p-1 is
  a prime number.
• 211 - 1  2047  23  89 is not prime and
  therefore p  11 does not yield a perfect number.
• In order for 2p - 1 to be prime, it is necessary
  but not sufficient that p should be prime.

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Marin Mersenne
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. Thus, there is a concrete
one-to-one association between even perfect
numbers and Mersenne primes.
Born September 8, 1588 in Oize in Maine, France.
He was a Jesuit educated Minim priest. Died
September 1, 1648 in Paris, France
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Para que un número de Mersenne sea primo,
necesariamente p debe ser primo. Pero esta
condición necesaria, lamentablemente no es
suficiente. El monje Marin Mersenne, padre de
estos números, hizo la atrevida afirmación en el
siglo XVII de que 267 - 1 era primo. Esta
conjetura fue discutida durante más de 250 años.
En 1903, Frank Nelson Cole, de la Universidad de
Columbia, dio una conferencia sobre el tema en
una reunión de la Sociedad Americana de
Matemáticas. Cole -que siempre fue un hombre de
pocas palabras- caminó hasta el pizarrón y, sin
decir nada, tomó la tiza y comenzó con la
aritmética que se usa para elevar 2 a la
sexagésima séptima potencia -cuenta Eric Temple
Bell, que estaba en el auditorio-. Entonces,
cuidadosamente, le restó 1, obteniendo
147.573.952.589.676.412.927. Sin una palabra pasó
a un espacio en blanco del pizarrón y multiplicó
a mano 193.707.721 por 761.838.257.287. Las dos
cuentas coincidían. La conjetura de Mersenne se
desvaneció en el limbo de la mitología
matemática. Por primera vez, que se recuerde, la
Asociación Nacional de Matemáticas aplaudió
vigorosamente al autor de un trabajo presentado
ante ella. Cole volvió a su asiento sin haber
pronunciado una sola palabra. Nadie le hizo
siquiera una pregunta. El mundo es un
pañuelo Computa, coopera y recicla aliens y
primos Bartolo Luque
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Mersenne primes
As of September 2009, only 47 Mersenne primes are
known, which means there are 47 perfect numbers
known, the largest being 243,112,608
(243,112,609 - 1) with 25,956,377 digits.
Mersenne prime
The search for new Mersenne primes is the goal of
the GIMPS distributed computing project.
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Mersenne primes
The first 39 even perfect numbers are
2p-1(2p - 1) for prime numbers 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.
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It is believed (but unproved) that this sequence
is infinite. The data suggests that the number
of terms up to exponent N is roughly K log N for
some constant K.
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• Theorem. x 2p-1(2p-1) is a perfect number when
  2p-1 is prime.
• Proof. For x to be a perfect number, it must be
  equal to the sum of its proper divisors. The
  divisors of 2p-1 are 1, 2, 22, ..., 2p-1. Since
  2p-1 is prime, its only divisors are 1 and
  itself.
• Therefore, the proper divisors of x are 1, 2,
  22, ..., 2p-1, 2(2p-1), 22(2p-1 ), ,
  2p-2(2p-1).

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• The sum of these divisors is
• ?i0p-1 2i (2p-1) ?i0p-2 2i 2p-1
  (2p-1) (2p-1 1) (2p-1) (1 2p-1 1)
  2p-1 (2p-1)
• Therefore, x2p-1 (2p-1) is a perfect number.
• Q.E.D.

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• Any even perfect number is the sum of the first
  natural numbers up to 2n-1.
• n 2 6 21 (22 -1) 123,
• n 3 28 22 (23 -1) 1234567,
• n 5 496 24 (25 -1) 12343031,
• n 7 8128 26 (27 -1) 123126127, etc.
• 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.

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• Any even perfect number (except 6) is the sum of
  the first 2(n-1)/2 odd cubes.
• 28 13 33,
• 496 13 33 53 73,
• 8128 13 33 53 73 93 113 133 153 ,
  etc.

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• The reciprocals of all positive factors of a
  perfect number add up to 2. Examples
• for 6
• for 28
  
• for 496
  

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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.
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Owing to their form, 2p-1(2p - 1), every even
perfect number is represented in binary as p 1s
followed by p - 1  0s. For example 610
1102 2810 111002 49610 1111100002
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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. http//oddperfect.org/pomer
ance.html 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.
Ore's harmonic number a positive integer whose
divisors have a harmonic mean that is an integer.
For example, the harmonic divisor number 6 has
the four divisors 1, 2, 3, and 6. Their harmonic
mean is an integer
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• Any odd perfect number N must satisfy the
  following conditions
• N gt 10300 (1989 R. P. Brent y G. L. Cohen
  demostraron que si existe un perfecto impar posee
  al menos 300 cifras).
• N is of the form
•                         
• where
• q, p1, ..., pk are distinct primes (Euler).
• q  a  1 (mod 4) (Euler).
• The smallest prime factor of N is less than
  (2k  8) / 3 (Grün 1952).
• Either qa gt 1020, or p j2ej  gt 1020 for some j
  (Cohen 1987).
• N lt 24k1 (Nielsen 2003).

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• 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).