Perfect numbers Números perfectos. s(n) = n s(6) = 1 + 2 + 3 = 6. Perfect numbers: Los pitagóricos fueron los primeros en preocuparse por ellos. San Agustín.

Presentación del tema: "Perfect numbers Números perfectos. s(n) = n s(6) = 1 + 2 + 3 = 6. Perfect numbers: Los pitagóricos fueron los primeros en preocuparse por ellos. San Agustín."— Transcripción de la presentación:

1 Perfect numbers Números perfectos

2 s(n) = n s(6) = 1 + 2 + 3 = 6. Perfect numbers: 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

3 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 OEISA000396OEIS

4 Euclid discovered that the first four perfect numbers are generated by the formula 2 p1 (2 p 1): for p = 2: 2 1 (2 2 1) = 6 for p = 3: 2 2 (2 3 1) = 28 for p = 5: 2 4 (2 5 1) = 496 for p = 7: 2 6 (2 7 1) = 8128 Noticing that 2 p 1 is a prime number in each instance. Euclid proved that the formula 2 p1 (2 p 1) gives a perfect number whenever 2 p 1 is prime. (Euclid, Prop. IX.36). Some Interesting Facts about Perfect Numbers

5 Euclid–Euler Theorem Euler proved that any even perfect number is given by the formula 2 p-1 (2 p -1), where 2 p -1 is a prime number. 2 11 1 = 2047 = 23 × 89 is not prime and therefore p = 11 does not yield a perfect number. In order for 2 p 1 to be prime, it is necessary but not sufficient that p should be prime.

6 Marin Mersenne Prime numbers of the form 2 p 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

7 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 2 67 - 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

8 As of September 2009, only 47 Mersenne primes are known, which means there are 47 perfect numbers known, the largest being: 2 43,112,608 × (2 43,112,609 1) with 25,956,377 digits. Mersenne primes The search for new Mersenne primes is the goal of the GIMPS distributed computing project. Mersenne prime

9 The first 39 even perfect numbers are 2 p1 (2 p 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).A000043OEIS 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. Mersenne primes

10 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.

11 Theorem. x = 2 p-1 (2 p -1) is a perfect number when 2 p -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 2 p-1 are 1, 2, 2 2,..., 2 p-1. Since 2 p -1 is prime, its only divisors are 1 and itself. Therefore, the proper divisors of x are 1, 2, 2 2,..., 2 p -1, 2(2 p -1), 2 2 (2 p -1 ), …, 2 p-2 (2 p -1).

12 The sum of these divisors is: i=0 p-1 2 i + (2 p -1) i=0 p-2 2 i = 2 p -1 + (2 p -1) (2 p-1 – 1) = (2 p -1) (1 + 2 p-1 – 1) = 2 p-1 (2 p -1) Therefore, x=2 p-1 (2 p -1) is a perfect number. Q.E.D.

13 Any even perfect number is the sum of the first natural numbers up to 2 n -1. n =2: 6 = 2 1 (2 2 -1) = 1+2+3, n =3: 28 = 2 2 (2 3 -1) = 1+2+3+4+5+6+7, n =5: 496 = 2 4 (2 5 -1) = 1+2+3+4+…+30+31, n =7: 8128 = 2 6 (2 7 -1) = 1+2+3+…+126+127, etc. Since any even perfect number has the form 2 p1 (2 p 1), it is the (2 p 1)th triangular number and the 2 p1 th hexagonal number. Like all triangular numbers, it is the sum of all natural numbers up to a certain point.

14 Any even perfect number (except 6) is the sum of the first 2 (n-1)/2 odd cubes. 28 = 1 3 + 3 3, 496 = 1 3 + 3 3 + 5 3 + 7 3, 8128 = 1 3 + 3 3 + 5 3 + 7 3 + 9 3 + 11 3 + 13 3 + 15 3, etc.

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

16 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.

17 Owing to their form, 2 p1 (2 p 1), every even perfect number is represented in binary as p 1s followed by p 1 0s. For example: 6 10 = 110 2 28 10 = 11100 2 496 10 = 111110000 2

18 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:

19 Any odd perfect number N must satisfy the following conditions: N > 10 300 (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, p 1,..., p k 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 q α > 10 20, or p j 2e j > 10 20 for some j (Cohen 1987). N < 2 4k+1 (Nielsen 2003).

20 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).