h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: Perfect Numbers - A2097632
Author: Gnomon - U151503

This topic was identified in the Challenge h2g2 forum as one not covered by the Guide, so I've written this entry. I hope it is reasonably well explained.

Hello Gnomon

I've never been famous for my mathmatical ability, (or any other ability come to that ) so I could just be the perfect numpty-person to read this entry.

However, I don't have time right now. I'll try to get back to you tomorrow.


F/b

How about the ability to offer a sympathetic ear? They don't examine that one in school, unfortunately.

I just read it through, and found it a bit hard going for my non mathmatical inklings... However, some of it rang a lot of bells about an article I read about this a few months back in some scientific journal yes, why am I as a non mathmatically minded individual reading articles in journals about mathmatics.... if we knew that I recon the odd perfect number would only be a matter of a few cigarettes away...
ahh, yes, back to the TIH.
It looks pretty good; I'm pressuming all the maths on adding up those numbers for that large perfect number you put on as a demonstration is correct, I couldn't add them up if I tried (and at least not reach the correct answer).
Good work

I really wrote this because someone said it was a topic that should be covered in the guide. I've done a lot of fiddling around with these numbers, but I'm not sure that they'd be of any interest to the general reader. If it isn't, I'll be perfectly happy to withdraw the entry, even though there is a "perfect-number-shaped" hole in the Guide.

Well, I've only come across them once before, as I said, if you were to ask me before I read your article, what one was.... I wouldn't have been able to say, and now I can!, Its certainly something that should be in the guide, lettuce see what the others think whereever they are (Did my previous post sound like that big a critisism? )

Hello everyone.

Well (as Gnomon knows) maths does not frighten me. In fact I found repeated use of the phrase "counting numbers" a bit grating, especially as there is A405352 to explain natural numbers.

For the more sophisticated reader, would it be worth mentioning the extent to which numbers are abundant or deficient, because if large numbers tended to be always deficient say, there would be good reason to suspect that perfect numbers might not exist (especially the odd ones).

Okay apart from these things though. It would be nice to have an entry on perfect numbers, so please disregard my remarks at whim.

Does the article link to other entrys on numbers? I can't remember, would be useful, where you mention other types of numbers in the text, if it could be linked to any articles on them...

Hey Gnomon I understood it all

Mind you, if somebody asks me about it in a couple of hours' time, I might be a bit stumped without referring to your entry first!

Now, what I want to know is..... why is it important for us to find the odd perfect number, if it exists of course? And if it doesn't, does it actually matter?




F/b

Hi Frenchbean! It isn't important at all. I mentioned that in the entry, but didn't make it clear. This is "pure mathematics" which basically means stuff that is true but nobody's found a use for. Much of mathematics has been considered "pure" until some engineer or technician found that it could be used for real world problems. Factorisation, an obscure topic that attracted eccentrics who wrote treatises in green ink, suddenly became hot stuff when some computer scientists designed a cryptography system based on it. Complex numbers (square root of minus one and all that) became useful when electrical engineers showed that they could be used for describing the phases of alternating current. Perfect numbers are something that no-one has yet found a use for. They may never find a use for them, in which case they will continue to be an obscure branch of maths which is pursued for its own sake.

I'll think about putting some of that into the entry.

Old Hairy, a good point about the relative frequency of abundant and deficient numbers. I don't know the answer. Do you?

Hi Gnomon

Thank you for confirming that there is no point

I think this is a great entry btw.


F/b

Hi Gnomon

I'm really too tired to take this all in at the moment, but a couple of things near the beginning (when I was more awake)

To found out which numbers are divisors > To find out which numbers are divisors

“You can stop searching at N-1;” Can’t you stop searching at ½N?

SK

Hello everyone.

Relative frequency of abundant/deficient numbers is apparently one of the great unknowns (but I didn't realise that when I mentioned it).

In trying to find out, I found out may things, summarised best in the very technical http://math.berkeley.edu/~jvoight/research/perfelem.pdf

Also found the rather interesting http://www.geocities.com/SoHo/Exhibit/8033/enlightened/pictureperfect.html which offer an alternative view of perfection

Much of the extensive literature on perfect numbers uses the term aliquot parts. I wonder if that should appear in this entry.

I dunno.

I rather liked this entry as it stands. It has a nice conversational tone and it all makes (some) sense to me and I'm about as nonmathematical as they come.

If you lot want to propose changes, I hope you don't ruin the tone as it currently stands.


- Jimi X

I stand firmly alongside Jimi X

Please don't make it too scarey


F/b

Another good one Gnomon.

I don't see any reason why an entry on a weird mathemathical subject cannot appear in the guide.

About the abundance of deficient of abundant numbers I only can say that there are infinite amounts of both.
Each prime is (almost) by definition deficient, and each n! abundant.
I strongly suspect there will be many more deficients than abundants.

Maybe interesting, the 40th mersenne prime is found a few weeks ago by GIMPS.
On November 17th GIMPS member Mxxx at Michigan State University proved 2^20996011-1 prime. At over 6.3 million digits it is easily the largest known prime number, beating GIMPS' 2 year old record of just over 4 million digits.


I leave it up to you to calculate the accompanying perfect number

Just for the record, I do not want it scary either.

Thank you, Old Scary!

An excellent entry Gnomon!

Tiny little typo at the end - things are sewn up rather than sown up.

Nice entry. Recreational mathematics is an area that needs its supporters. Keep up the good work.