# More Perfect Numbers **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [July 1, 2005, 1:14pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676 "2005-07-01T13:14:53Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 3, 2005, 4:18am UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/21 "2005-07-03T04:18:30Z")
I can see where that would be confusing. For 2[sup]n - 1[/sup] and 2[sup]n[/sup] - 1, the superscript represents exponentiation. For 10[sup]n - 2[/sup] and 1[sup]n - 1[/sup], the superscript represents concatenation. ---
**Author:** ![Plynck](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Plynck](https://boards.straightdope.com/u/Plynck)\ **Post date:** [July 5, 2005, 1:43pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/22 "2005-07-05T13:43:12Z")
Just curious. The report refers to the first six perfect numbers, starting with 6. Why isn’t 1 considered to be a perfect number? ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [July 5, 2005, 1:46pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/23 "2005-07-05T13:46:00Z")
> [@Plynck](#): > > Just curious. > > The report refers to the first six perfect numbers, starting with 6. > > Why isn’t 1 considered to be a perfect number? The sum of the factors does not include the number itself. 6=3+2+1 ---
**Author:** ![Plynck](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Plynck](https://boards.straightdope.com/u/Plynck)\ **Post date:** [July 5, 2005, 1:53pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/24 "2005-07-05T13:53:38Z")
> [@RM Mentock](#): > > The sum of the factors does not include the number itself. 6=3+2+1 Ah, completely missed that… Thanks. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 5, 2005, 3:26pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/25 "2005-07-05T15:26:40Z")
> [@Plynck](#): > > Just curious. > > The report refers to the first six perfect numbers, starting with 6. > > Why isn’t 1 considered to be a perfect number? Also, there’s no corresponding Mersenne prime, so if 1 were a perfect number, it would have to be a special case. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [July 5, 2005, 5:57pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/26 "2005-07-05T17:57:24Z")
> [@ultrafilter](#): > > Also, there’s no corresponding Mersenne prime, so if 1 were a perfect number, it would have to be a special case. Perfect numbers are not defined by using the Mersenne primes. Odd perfect numbers, if they do exist, would not have Mersenne primes associated with them either. ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [July 5, 2005, 6:32pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/27 "2005-07-05T18:32:29Z")
> [@RM Mentock](#): > > Perfect numbers are not defined by using the Mersenne primes. Odd perfect numbers, if they do exist, would not have Mersenne primes associated with them either. Moreover, it does follow the 2[sup]n-1[/sup] \* 2[sup]n[/sup]-1 formula, for n=1, counting 2[sup]1[/sup]-1 (1) as a degenerate quasi-prime. ---
**Author:** ![Hypnagogic\_Jerk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hypnagogic_jerk/32/4252_2.png) [@Hypnagogic\_Jerk](https://boards.straightdope.com/u/Hypnagogic_Jerk)\ **Post date:** [July 5, 2005, 11:18pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/28 "2005-07-05T23:18:15Z")
> [@Mathochist](#): > > Actually, now that I’ve looked at the final result of my work, it seems that the joke was so carefully crafted and hidden that it’s been edited out! I’m curious, and surprised that no one has asked yet: what was your joke? ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [July 6, 2005, 6:28am UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/29 "2005-07-06T06:28:38Z")
> [@severus](#): > > I’m curious, and surprised that no one has asked yet: what was your joke? It should have read “Descartes _had_ studied pluperfect numbers”. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [July 6, 2005, 6:31am UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/30 "2005-07-06T06:31:00Z")
> [@John W. Kennedy](#): > > Moreover, it does follow the 2[sup]n-1[/sup] \* 2[sup]n[/sup]-1 formula, for n=1, counting 2[sup]1[/sup]-1 (1) as a degenerate quasi-prime. (1) in any unital ring cannot be a prime ideal because it is the entirety of the ring. Then R/(1) is the degenerate ring, which is not an integral domain. A prime ideal of R is one such that R/I is an integral domain. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [July 7, 2005, 4:55pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/31 "2005-07-07T16:55:07Z")
The On-Line Encyclopedia of Integer Sequences link is no longer there (it didn’t work when I tried it before), but your list of the first nine perfect numbers is missing the 6th, 8,589,869,056 ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 7, 2005, 5:00pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/32 "2005-07-07T17:00:13Z")
Hey, no fair trying to sneak a grammar joke into a math report! ---
**Author:** ![Excalibre](https://avatars.discourse-cdn.com/v4/letter/e/898d66/32.png) [@Excalibre](https://boards.straightdope.com/u/Excalibre)\ **Post date:** [July 7, 2005, 8:41pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/33 "2005-07-07T20:41:20Z")
> [@Mathochist](#): > > (1) in any unital ring cannot be a prime ideal because it is the entirety of the ring. Then R/(1) is the degenerate ring, which is not an integral domain. A prime ideal of R is one such that R/I is an integral domain. Thanks for clearing that up! :: goes off to consider a program that helps teach English to mathematicians :: ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 7, 2005, 8:43pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/34 "2005-07-07T20:43:27Z")
> [@Excalibre](#): > > Thanks for clearing that up! > > :: goes off to consider a program that helps teach English to mathematicians :: Makes perfect sense to me. ---
**Author:** ![Excalibre](https://avatars.discourse-cdn.com/v4/letter/e/898d66/32.png) [@Excalibre](https://boards.straightdope.com/u/Excalibre)\ **Post date:** [July 7, 2005, 9:46pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/35 "2005-07-07T21:46:27Z")
> [@ultrafilter](#): > > Makes perfect sense to me. I didn’t say it wasn’t communication. I said it wasn’t English. 🙂 ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [July 8, 2005, 10:44pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/36 "2005-07-08T22:44:25Z")
> [@Excalibre](#): > > Thanks for clearing that up! > > :: goes off to consider a program that helps teach English to mathematicians :: Of course it’s not English. It’s jargon. You expect mathematicians to use strictly nontechnical vocabulary when discussing carefully-defined technical terms? Hell, why don’t you tell physicists to call quarks “thingies”? ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [July 9, 2005, 7:03pm UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/37 "2005-07-09T19:03:55Z")
> [@Mathochist](#): > > Of course it’s not English. It’s jargon. You expect mathematicians to use strictly nontechnical vocabulary when discussing carefully-defined technical terms? Hell, why don’t you tell physicists to call quarks “thingies”? That’s not fair. The name “quark” is essentially “thingie” 🙂 But that’s an interesting issue. Can you make jargon accessible to the layperson? Dex thinks so. There are a lot of advantages to it. As Feynman used to say, if you can’t present a concept in the terms of a freshman lecture, you really don’t understand it. Sometimes, just the exercise of trying to express it in simpler terms makes the concepts more clear. Your remark, which seemed to be aimed at John’s calling 1 a “quasi-prime”, was sorta the equivalent of saying, “most mathematicians don’t consider 1 to be a prime” but that is pretty much what John is saying to begin with. ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [July 10, 2005, 3:34am UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/38 "2005-07-10T03:34:16Z")
There are a number of reasons that 1 is not considered a prime, traditionally, that if 1 were considered a prime, there would be a good many statements that now say “All primes…” that would have to be emended to “All primes except 1…” Remember, a certain amount of math can be regarded as always true, whether we like it or not, but some things, like the word “prime”, are defined for the convenience of the human mathematicians who use them. At any rate, you can regard 1 as a kinda-sorta prime, and as a kinda-sorta perfect number, too, and it is interesting that the two mesh with each other, in that 2[sup]0[/sup]\*(2[sup]1[/sup]-1) = 1 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 10, 2005, 3:47am UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/39 "2005-07-10T03:47:18Z")
> [@John W. Kennedy](#): > > Remember, a certain amount of math can be regarded as always true, whether we like it or not, but some things, like the word “prime”, are defined for the convenience of the human mathematicians who use them. I get what you’re trying to say, but I feel I should point out that there’s nothing you can prove with this definition of “prime” that you can’t prove without it. My old logic textbook has a section on definitions that shows that in general, they’re all conveniences and don’t change anything. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [July 10, 2005, 5:43am UTC](https://boards.straightdope.com/t/more-perfect-numbers/310676/40 "2005-07-10T05:43:02Z")
> [@John W. Kennedy](#): > > At any rate, you can regard 1 as a kinda-sorta prime This may “kinda-sorta” work for primes in the ring of integers, but as I just pointed out this doesn’t generalize. The definition of primes most people know is equivalent to the real definition only under certain conditions, and is more easily stated and understood. [Previous page](https://boards.straightdope.com/t/more-perfect-numbers/310676.md?page=1) [Next page](https://boards.straightdope.com/t/more-perfect-numbers/310676.md?page=3)