mersenneforum.org  

Go Back   mersenneforum.org > Factoring Projects > Aliquot Sequences

Reply
 
Thread Tools
Old 2015-11-22, 10:03   #1
Dubslow
Basketry That Evening!
 
Dubslow's Avatar
 
"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3·2,399 Posts
Arrow Introduction to Aliquot Analysis

In the last few weeks I've finally come to fully understand Clifford Stern's work -- it took me quite a lot of effort of working out a lot of significant details that Stern's page seems to otherwise assume that you already know. And the other thread I created recently had other related questions I was asking myself and have worked out since, with hints and help from the forum members.

Having in fact worked out all the details, in retrospect it would have been nice if someone presented the missing details along with the stuff that Stern was talking about.

So that's what I've done. I've made a large new theory page that starts from the very basics, defining what an aliquot sequence is etc, and working its way all the way up through the entirety of Clifford Stern's definitions and conclusions, except presented in a pedagogically oriented way (I hope). That is, all the definitions are properly motivated, as opposed to saying "here's a definition, and here's why it's important" as if some math god knew ahead of time why it's important and where to look to solve such matters. (I've also introduced a bit of new notation that helped simplify my understanding of things.)

Although the page includes almost no theory beyond Clifford's material, some here may yet find it educational, and I'd appreciate it if even the experts read it and gave me feedback. (Yes, the formatting is awful with all inline equations, and yes there could easily be a hundred links in the page, but I've only got so much time each day. I'm more interested in pedagogical feedback )

Also there's a small side problem which I've been unable to resolve, and everyone's thoughts on the matter are quite welcome (especially you experts please!)

Edit: Augh! The github preview thingy doesn't allow the javascript, which is kinda important to render the math... The preview links use HTTPS (as they should) so my browser blocks the http-not-s link to MathJax JS in the page. If the math doesn't render for you, overwrite your browser on that matter until the changes are pulled to rechenkraft

The main page: https://htmlpreview.github.io/?https...-analysis.html (soon to be available here when the rechencraft server pulls the latest commits)

And the small side problem with which I would appreciate some external insight: https://htmlpreview.github.io/?https...imepowers.html (again soon to be here)

Last fiddled with by Dubslow on 2015-11-22 at 10:11
Dubslow is offline   Reply With Quote
Old 2015-11-28, 10:40   #2
davar55
 
davar55's Avatar
 
May 2004
New York City

2·2,099 Posts
Default

Just starting to read your refreshing work.
Two paragraphs in and it looks and reads clean.
davar55 is offline   Reply With Quote
Old 2015-11-28, 11:04   #3
davar55
 
davar55's Avatar
 
May 2004
New York City

2×2,099 Posts
Default

In section 2:

where you have

Code:
prod of a-sub-i + 1
it should perhaps read

Code:
prod of (a-sub-i + 1)
Do you want or need this type of comment for minor points or typos?
davar55 is offline   Reply With Quote
Old 2015-11-28, 11:17   #4
Dubslow
Basketry That Evening!
 
Dubslow's Avatar
 
"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

11100000111012 Posts
Default

Quote:
Originally Posted by davar55 View Post
Just starting to read your refreshing work.
Two paragraphs in and it looks and reads clean.
Note that the rechenkraft version is currently not quite up to date with some typo fixes, a couple of incorrect statements removed, and of course the lack of the proof axn provided. Use this for now.

Quote:
Originally Posted by davar55 View Post
In section 2:

where you have

Code:
prod of a-sub-i + 1
it should perhaps read

Code:
prod of (a-sub-i + 1)
Do you want or need this type of comment for minor points or typos?

Please. Though perhaps the thread in the aliquot forum is more appropriate?

Last fiddled with by Dubslow on 2015-11-28 at 11:38
Dubslow is offline   Reply With Quote
Old 2015-11-28, 12:01   #5
davar55
 
davar55's Avatar
 
May 2004
New York City

10000011001102 Posts
Default

Quote:
Originally Posted by Dubslow View Post
Note that the rechenkraft version is currently not quite up to date with some typo fixes, a couple of incorrect statements removed, and of course the lack of the proof axn provided. Use this for now.
...
Please. Though perhaps the thread in the aliquot forum is more appropriate?
That is where I still see the missing parentheses.

I just happened on your post here, and so started reading your aliquot paper version, and found this
possible typo. I may look into more later.
davar55 is offline   Reply With Quote
Old 2015-11-28, 18:53   #6
davar55
 
davar55's Avatar
 
May 2004
New York City

2·2,099 Posts
Default

Quote:
Originally Posted by davar55 View Post
That is where I still see the missing parentheses.
I just happened on your post here, and so started reading your aliquot paper version, and found this
possible typo. I may look into more later.
I should have asked you whether you caught the typo already.
The document is yours, of course.

The next typo is in (p-to-the-a minus 1) / (p - 1) [ you wrote b for a ].

I'll stop now. Wish you success.

Last fiddled with by davar55 on 2015-11-28 at 18:57
davar55 is offline   Reply With Quote
Old 2015-11-29, 05:06   #7
Dubslow
Basketry That Evening!
 
Dubslow's Avatar
 
"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

719710 Posts
Default

You still see missing parens? Where?

The a->b typo's been fixed
Dubslow is offline   Reply With Quote
Old 2015-11-29, 11:43   #8
Happy5214
 
Happy5214's Avatar
 
"Alexander"
Nov 2008
The Alamo City

22×97 Posts
Arrow Introduction to Aliquot Analysis

Quote:
Originally Posted by Dubslow View Post
Happy, your analysis is indeed correct. Though I am curious, how did you come up with the twos count of prime powers? The proof isn't exactly trivial.
Luck? As I said, I'm not a math expert. I'm actually a college student with a CS major. IIRC, I just used Python to calculate a bunch of 2s counts for prime powers and noticed the pattern I described. Thanks for the validation. It was just time and pattern recognition on my part. No rigorous mathematical proofs involved. Not very impressive, but it still worked.
Happy5214 is offline   Reply With Quote
Old 2015-11-29, 12:23   #9
Dubslow
Basketry That Evening!
 
Dubslow's Avatar
 
"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

11100000111012 Posts
Default

Quote:
Originally Posted by Happy5214 View Post
Luck? As I said, I'm not a math expert. I'm actually a college student with a CS major. IIRC, I just used Python to calculate a bunch of 2s counts for prime powers and noticed the pattern I described. Thanks for the validation. It was just time and pattern recognition on my part. No rigorous mathematical proofs involved. Not very impressive, but it still worked.
That's actually exactly what I did!

http://htmlpreview.github.io/?http:/...imepowers.html

"But in fact, in my numerical tests, I have been unable to find a counter example where τ(p^a)≠τ(p)+β(l) [where a=2l-1], where the search extended to p<10^9, and each prime was tested up to a=99."

^ said tests were run in Python, and they're what made me look for the proof at all (which axn provided ). In fact, all code I've ever written for aliquot sequences, including the website, is written in Python.

Last fiddled with by Dubslow on 2015-11-29 at 12:24
Dubslow is offline   Reply With Quote
Old 2015-11-29, 13:54   #10
davar55
 
davar55's Avatar
 
May 2004
New York City

2×2,099 Posts
Default

Quote:
Originally Posted by Dubslow View Post
You still see missing parens? Where?
The a->b typo's been fixed
Both fixed.
davar55 is offline   Reply With Quote
Old 2015-11-29, 22:44   #11
henryzz
Just call me Henry
 
henryzz's Avatar
 
"David"
Sep 2007
Cambridge (GMT/BST)

166416 Posts
Default

I am slowly working through your page. It looks very good so far.
Thought I would point out another form of guide:

Take 2^8*7*73 for example \sigma(73)=74=2*37. This means that 37 | a(2^8*7*73*37*n). This process can be repeated to get 2^8*7*73*37*19*5*3*n which can't loose 7, 73, 37, 19, 5 or 3 as a factor. These can be fairly vulnerable to getting squared terms and then loosing the guide but they are fairly stable and are highly abundant.
Here is a fairly good example:
http://factordb.com/sequences.php?se...nge&fr=0&to=30


Can  2^3*3*5*n mutate directly to the form 2^3*3^2*5*n? It rings a bell that there are restrictions on how 2^3*3*5*n can mutate. Forgive me if this is mentioned. I have only properly read up to section 3 and glanced at the rest.
henryzz is online now   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
An introduction to quantum computing, without the physics only_human Other Mathematical Topics 5 2017-10-17 19:55
Self introduction 0x6f Lounge 9 2017-05-28 21:36
Introduction to Algebraic and Abelian Functions bearnol Miscellaneous Math 2 2015-12-30 05:32
MOOC: Introduction to Mathematical Thinking wblipp Lounge 2 2014-08-29 06:20
Comments on ElevenSmooth Introduction, Please wblipp ElevenSmooth 0 2003-11-24 04:55

All times are UTC. The time now is 21:50.

Mon Oct 26 21:50:55 UTC 2020 up 46 days, 19:01, 0 users, load averages: 1.58, 1.64, 1.74

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.