Perfect number

A perfect number ( other Greek ἀριθμὸς τέλειος ) is a natural number equal to the sum of all its own divisors (that is, all positive divisors other than the number itself). As natural numbers increase, perfect numbers are less and less common. It is not known whether the set of all perfect numbers is infinite.

Perfect numbers form the sequence ^[1] :

6 ,

28 ,

496 ,

8128 ,

33 550 336 ,

8 589 869 056 ,

137 438 691 328 ,

2 305 843 008 139 952 128 ,

2 658 455 991 569 831 744 654 692 615 953 842 176 ,

191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216 , ...

Content

Examples

1st perfect number - 6 has the following own divisors: 1, 2, 3; their sum is 6.
2nd perfect number - 28 has the following own divisors: 1, 2, 4, 7, 14; their total is 28.
3rd perfect number - 496 has the following proper divisors: 1, 2, 4, 8, 16, 31, 62, 124, 248; their total is 496.
4th perfect number - 8128 has the following own divisors: 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064; their sum is 8128.

Study History

Perfect Even Numbers

The algorithm for constructing even perfect numbers is described in the IX book of the Beginning of Euclid , where it was proved that the number $\ 2^{p-1}(2^{p}-1)$ ${\ displaystyle \ 2 ^ {p-1} (2 ^ {p} -1)}$ is perfect if the number $\ 2^{p}-1$ ${\ displaystyle \ 2 ^ {p} -1}$ is prime (the so-called prime Mersenne numbers ) ^[2] . Subsequently, Leonard Euler proved that all even perfect numbers have the form indicated by Euclid.

The first four perfect numbers (corresponding to p = 2, 3, 5, and 7) are given in the Nikomakh Gerazsky Arithmetic . The fifth perfect number 33 550 336 , corresponding to p = 13, was discovered by the German mathematician Regiomontan ( XV century ). In the XVI century, the German scientist Scheibel found two more perfect numbers: 8 589 869 056 and 137 438 691 328 . They correspond to p = 17 and p = 19. At the beginning of the XX century, three more perfect numbers were found (for p = 89, 107 and 127). In the future, the search was slowed down until the middle of the 20th century, when with the advent of computers, computations that exceeded human capabilities became possible.

For 2019, 51 perfect numbers are known, arising from Mersenne primes , which the GIMPS distributed computing project is looking for.

Odd perfect numbers

Odd perfect numbers have not yet been discovered, but it has not been proved that they do not exist. It is also unknown whether the set of odd perfect numbers is finite, if they exist.

It is proved that an odd perfect number, if it exists, exceeds 10 ¹⁵⁰⁰ ; the number of prime divisors of such a number, taking into account the multiplicity, is not less than 101 ^[3] . OddPerfect.org distributed computing project is looking for odd perfect numbers.

Properties

All even perfect numbers (except 6) are the sum of the cubes of consecutive odd positive integers

1^{3}+3^{3}+5^{3}+\ldots +(2n-1)^{3}=n^{2}(2n^{2}-1)

{\ displaystyle 1 ^ {3} + 3 ^ {3} + 5 ^ {3} + \ ldots + (2n-1) ^ {3} = n ^ {2} (2n ^ {2} -1)}

All even perfect numbers are triangular numbers ; in addition, they are hexagonal numbers , that is, can be represented as $n({2n-1})$ ${\ displaystyle n ({2n-1})}$ for some natural number $n$ ${\ displaystyle n}$ .
The sum of all numbers inverse to the divisors of a perfect number (including the number itself) is 2. This is a direct consequence of the definition and the fact that the sum of the divisors when divided by the number itself gives the sum of the numbers inverse to the dividers.
All perfect numbers are Ore numbers .
All even perfect numbers, except 6 and 496, end in decimal notation on 16, 28, 36, 56 or 76.
All even perfect binary numbers contain first $p$ ${\ displaystyle p}$ units followed by $p-1$ ${\ displaystyle p-1}$ zeros (a consequence of their general presentation).
If you add up all the digits of an even perfect number (except 6), then add up all the digits of the resulting number and repeat this way until you get a single-digit number ^[4] , then this number will be 1 (2 + 8 = 10, 1 + 0 = 1; 4 + 9 + 6 = 19, 1 + 9 = 10 ...) Equivalent wording: the remainder of dividing an even perfect number other than 6 by 9 is 1.

Interesting Facts

The special ("perfect") nature of the numbers 6 and 28 was recognized in cultures based on the Abrahamic religions , claiming that God created the world in 6 days and drawing attention to the fact that the moon rotates around the earth in about 28 days.

James A. Eshelman in the book "The Jewish hierarchical names of Beria" ^[5] writes that in accordance with gematria :

No less important is the idea expressed by the number 496. This is the “Theosophical extension” of the number 31 (that is, the sum of all integers from 1 to 31). Among other things, this is the sum of the word Malchut (kingdom). Thus, the Kingdom, the full manifestation of the primary idea of God, appears in gematria as a natural complement or manifestation of the number 31, which is the number of the name 78.

“ Leviathan ” (lit. “wriggling”) is one of the four Princes of Darkness, embodied in the form of a serpent. Therefore, to keep Leviathan is to control the energies of Nefesh associated with the Sefira Yesod. Secondly, “bending serpent” can also mean “coiled serpent rings,” that is, Kundalini . Thirdly, the gematria of the word “Leviathan” is 496, as well as the words “Malchut” (Kingdom); The notion that the archangel Yesod restrains the nature of Malchut provides rich food for thought. Fourth, the number 496 is the sum of the numbers from 1 to 31, that is, the full expansion, or manifestation, of the name "El", the divine name of the three highest Sefirot in Beria (including the Sefira Keter , whose angel is Jehoel).

In the work “City of God,” St. Augustine wrote ^[6] :

The number 6 is completely in itself, and not because the Lord created everything in 6 days; rather, on the contrary, God created everything in 6 days because this number is perfect. And it would remain perfect, even if there were no creation in 6 days.

Notes

↑ Sequence A000396 in OEIS : perfect numbers = Perfect numbers n: n is equal to the sum of the proper divisors of n
↑ Perfect Beauty and Perfect Uselessness of Perfect Numbers
↑ Ochem, Pascal; Rao, Michaël. Odd perfect numbers are greater than 10 ¹⁵⁰⁰ (English) // Mathematics of Computation : journal. - 2012. - Vol. 81 , no. 279 . - P. 1869-1877 . - ISSN 0025-5718 . - DOI : 10.1090 / S0025-5718-2012-02563-4 .
↑ see Numerology # Reduction of numbers to numbers
↑ Numbers
↑ Simon Singh . Great Farm Theorem. with. 9 (inaccessible link) .

Links

Depman I. Perfect numbers // Quantum . - 1991. - No. 5 . - S. 13-17 .
Evgeny Epifanov. Perfect numbers (neopr.) . Elements

[1] Sequence A000396 in OEIS : perfect numbers = Perfect numbers n: n is equal to the sum of the proper divisors of n

[2] Perfect Beauty and Perfect Uselessness of Perfect Numbers

[Ochem_and_Rao_(2012)-3] Ochem, Pascal; Rao, Michaël. Odd perfect numbers are greater than 10 ¹⁵⁰⁰ (English) // Mathematics of Computation : journal. - 2012. - Vol. 81 , no. 279 . - P. 1869-1877 . - ISSN 0025-5718 . - DOI : 10.1090 / S0025-5718-2012-02563-4 .

[4] see Numerology # Reduction of numbers to numbers

[5] Numbers

[6] Simon Singh . Great Farm Theorem. with. 9 (inaccessible link) .