Съвършено число: Разлика между версии
Изтрито е съдържание Добавено е съдържание
Ред 10:
== Пресмятане на съвършени числа ==
Още [[Евклид]] в своите "Елементи" е установил, че първите четири съвършени числа могат да се пресметнат по формулата
: <math>2^{n-1}(2^n - 1)</math>,
където <math>2^n - 1</math> е просто число.
* За ''n'' = 2: <math>2^1(2^2-1)</math> = 6 = 1 + 2 + 3
Ред 20:
* За ''n'' = 5: <math>2^4(2^5-1)</math> = 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
* За ''n'' = 7: <math>2^6(2^7-1)</math> = 8 128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064
Повече от хилядолетие след Евклид Ибн ал-Хайтам (Алхазен)
Over a millennium after Euclid, Ibn al-Haytham (Alhazen) circa 1000 AD realized that every even perfect number is of the form 2n−1(2n − 1) where 2n − 1 is prime, but he was not able to prove this result.[1] It was not until the 18th century that Leonhard Euler proved that the formula 2n−1(2n − 1) will yield all the even perfect numbers. Thus, 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 September 2007, only 44 Mersenne primes are known,[2] which means there are 44 perfect numbers known, the largest being 232,582,656 × (232,582,657 − 1) with 19,616,714 digits.
The first 39 even perfect numbers are 2n−1(2n − 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. 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 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, \,
Всички познати съвършени числа < 10<sup>18</sup> са четни. Не е известно дали има нечетно съвършено число.
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