Transcript Diapositiva 1

Perfect numbers
Números perfectos
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
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
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:
for p = 3:
for p = 5:
for p = 7:
21(22 − 1) = 6
22(23 − 1) = 28
24(25 − 1) = 496
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).
Euclid–Euler 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.
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
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
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.
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.
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.
• 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).
•
The sum of these divisors is:
i=0p-1 2i + (2p-1) i=0p-2 2i
= 2p-1 + (2p-1) (2p-1 – 1)
= (2p-1) (1 + 2p-1 – 1)
= 2p-1 (2p-1)
Therefore, x=2p-1 (2p-1) is a perfect number.
Q.E.D.
Any even perfect number is the sum of the
first natural numbers up to 2n-1.
n =2:
6 = 21 (22 -1) = 1+2+3,
n =3:
28 = 22 (23 -1) = 1+2+3+4+5+6+7,
n =5: 496 = 24 (25 -1) = 1+2+3+4+…+30+31,
n =7: 8128 = 26 (27 -1) = 1+2+3+…+126+127,
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.
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.
The reciprocals of all positive factors of a
perfect number add up to 2. Examples:
for 6:
1 1 1 1
   2
6 3 2 1
;
1
1 1 1 1 1
     2
for 28:
28 14 7 4 2 1
;
1
1
1
1
1 1 1 1 1 1



      2
for 496:
496 248 124 62 31 16 8 4 2 1
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.
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
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/pomerance.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:
Any odd perfect number N must satisfy the
following conditions:
•N > 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 ≡ 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).
•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).