# Perfect numbers **URL:** \ **Category:** Factual Questions\ **Created:** [December 13, 2007, 4:12am UTC](https://boards.straightdope.com/t/perfect-numbers/429816 "2007-12-13T04:12:18Z")\ **Posts on this page:** 10\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [December 13, 2007, 4:12am UTC](https://boards.straightdope.com/t/perfect-numbers/429816/1 "2007-12-13T04:12:18Z")
A perfect number is one that is the sum of its divisors. > **[Perfect number](https://en.wikipedia.org/wiki/Perfect_number)** > > In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. > The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including it... An interesting characteristic of perfect numbers, in decimal notation, is every known one ends in 6 or 8. I have noticed something about these numbers in binary: 6 \> 110 28 \> 11100 496 \> 111110000 8128 \> 1111111000000 2096128 \> 111111111110000000000 In short, the Euler Generator…2\[1\](2\[2\]-1) yields n number of ones followed by n-1 number of zeroes. Anything else come up in other notations, such as heximal or octal? * * * 1. n-1 2. n ---
**Author:** ![ArmenE](https://avatars.discourse-cdn.com/v4/letter/a/e9bcb4/32.png) [@ArmenE](https://boards.straightdope.com/u/ArmenE)\ **Post date:** [December 13, 2007, 5:11am UTC](https://boards.straightdope.com/t/perfect-numbers/429816/2 "2007-12-13T05:11:26Z")
[QUOTE=Enola Straight] A perfect number is one that is the sum of its divisors. > **[Perfect number](https://en.wikipedia.org/wiki/Perfect_number)** > > In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2, and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, because 28 has proper divisors 1, 2, 4, 7, 14, and 1 + 2 + 4 + 7 + 14 = 28. > The first seven perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, and 137438691328. > The sum of proper divisors of a number is... An interesting characteristic of perfect numbers, in decimal notation, is every known one ends in 6 or 8. I have noticed something about these numbers in binary: 6 \> 110 28 \> 11100 496 \> 111110000 8128 \> 1111111000000 2096128 \> 111111111110000000000 In short, the Euler Generator…2\[1\](2\[2\]-1) yields n number of ones followed by n-1 number of zeroes. Anything else come up in other notations, such as heximal or octal? [/QUOTE] 2096128 isn’t a perfect number * * * 1. n-1 2. n ---
**Author:** ![J\_Cubed](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@J\_Cubed](https://boards.straightdope.com/u/J_Cubed)\ **Post date:** [December 13, 2007, 5:17am UTC](https://boards.straightdope.com/t/perfect-numbers/429816/3 "2007-12-13T05:17:20Z")
Cool. Hexadecimal perfects only contain 0, 1, C, F. Octals beyond 34 only contain 0, 1, 6, 7. Perfect numbers in base infinity show no pattern. ```auto Octal Decimal Hexadecimal 6 6 6 34 28 1C 760 496 1F0 17700 8128 1FC0 177770000 33550336 1FFF000 77777600000 8589869056 1FFFF0000 1777777000000 137438691328 1FFFFC0000 177777777770000000000 2305843008139952128 1FFFFFFFC0000000 ``` (To the limit of MS Excel and Calculator) ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [December 13, 2007, 5:24am UTC](https://boards.straightdope.com/t/perfect-numbers/429816/4 "2007-12-13T05:24:04Z")
[QUOTE=Enola Straight] A perfect number is one that is the sum of its divisors. > **[Perfect number](https://en.wikipedia.org/wiki/Perfect_number)** > > In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2, and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, because 28 has proper divisors 1, 2, 4, 7, 14, and 1 + 2 + 4 + 7 + 14 = 28. > The first seven perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, and 137438691328. > The sum of proper divisors of a number is... An interesting characteristic of perfect numbers, in decimal notation, is every known one ends in 6 or 8. I have noticed something about these numbers in binary: 6 \> 110 28 \> 11100 496 \> 111110000 8128 \> 1111111000000 2096128 \> 111111111110000000000 In short, the Euler Generator…2\[1\](2\[2\]-1) yields n number of ones followed by n-1 number of zeroes. Anything else come up in other notations, such as heximal or octal? [/QUOTE] This is a natual consequence of a basic theorem of perfect numbers i.e. if (2^n)-1 is prime then (2^(n-1)) x ((2^n)-1) is perfect. * * * 1. n-1 2. n ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [December 13, 2007, 4:53pm UTC](https://boards.straightdope.com/t/perfect-numbers/429816/5 "2007-12-13T16:53:33Z")
[QUOTE=ArmenE] 2096128 isn’t a perfect number [/QUOTE] Uh, sorry. I thought n=11 did generate perfect. Anyway, Not hexidecimal…heximal: base six. ---
**Author:** ![Klepto21](https://avatars.discourse-cdn.com/v4/letter/k/65b543/32.png) [@Klepto21](https://boards.straightdope.com/u/Klepto21)\ **Post date:** [December 13, 2007, 8:47pm UTC](https://boards.straightdope.com/t/perfect-numbers/429816/6 "2007-12-13T20:47:06Z")
[QUOTE=J Cubed] Perfect numbers in base infinity show no pattern. [/QUOTE] That’s a joke, right? ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 13, 2007, 9:22pm UTC](https://boards.straightdope.com/t/perfect-numbers/429816/7 "2007-12-13T21:22:56Z")
[QUOTE=Klepto21] That’s a joke, right? [/QUOTE] I’m no math wiz by any means, but I would assume that in base infinity there could never be any pattern because every number would be a unique digit. ---
**Author:** ![Klepto21](https://avatars.discourse-cdn.com/v4/letter/k/65b543/32.png) [@Klepto21](https://boards.straightdope.com/u/Klepto21)\ **Post date:** [December 13, 2007, 9:27pm UTC](https://boards.straightdope.com/t/perfect-numbers/429816/8 "2007-12-13T21:27:31Z")
[QUOTE=Liberal] I’m no math wiz by any means, but I would assume that in base infinity there could never be any pattern because every number would be a unique digit. [/QUOTE] I’m no whiz either, which is why I asked. But yeah, there could never be a pattern. _now that I’m sure it’s a joke:_ “Base Infinity”: HA! ---
**Author:** ![Bootis](https://avatars.discourse-cdn.com/v4/letter/b/e8c25b/32.png) [@Bootis](https://boards.straightdope.com/u/Bootis)\ **Post date:** [December 14, 2007, 8:25am UTC](https://boards.straightdope.com/t/perfect-numbers/429816/9 "2007-12-14T08:25:46Z")
Is there any application for perfect numbers other than being notable? Or for instance, is the formula used to calculate perfect numbers used for anything in physics or mathematics, other than soley calculating perfect numbers? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 14, 2007, 8:39am UTC](https://boards.straightdope.com/t/perfect-numbers/429816/10 "2007-12-14T08:39:37Z")
[QUOTE=Saint Cad] This is a natual consequence of a basic theorem of perfect numbers i.e. if (2^n)-1 is prime then (2^(n-1)) x ((2^n)-1) is perfect. [/QUOTE] Though that’s a true statement, it seems that “Every even perfect number’s last digit is 6 or 8” is rather a consequence of a different true statement, somewhat converse: every even perfect number is of the form (2^(n-1)) x ((2^n)-1) where (2^n) - 1 is prime.