Introduction to Aliquot Analysis

Dubslow · 2015-11-22, 10:03

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)

davar55 · 2015-11-28, 10:40

Just starting to read your refreshing work.
Two paragraphs in and it looks and reads clean.

davar55 · 2015-11-28, 11:04

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?

Dubslow · 2015-11-28, 11:17

Quote:

Originally Posted by davar55

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

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?

davar55 · 2015-11-28, 12:01

Quote:

Originally Posted by Dubslow

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 · 2015-11-28, 18:53

Quote:

Originally Posted by davar55

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.

Dubslow · 2015-11-29, 05:06

You still see missing parens? Where?

The a->b typo's been fixed

Happy5214 · 2015-11-29, 11:43

Quote:

Originally Posted by Dubslow

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.

Dubslow · 2015-11-29, 12:23

Quote:

Originally Posted by Happy5214

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.

davar55 · 2015-11-29, 13:54

Quote:

Originally Posted by Dubslow

You still see missing parens? Where?
The a->b typo's been fixed

Both fixed.

henryzz · 2015-11-29, 22:44

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.

2015-11-22, 10:03	#1
Dubslow Basketry That Evening! "Bunslow the Bold" Jun 2011 40<A<43 -89<O<-88 3·29·83 Posts	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

2015-11-28, 11:04	#3
davar55 May 2004 New York City 4235₁₀ Posts	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?

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2015-11-28, 10:40	#2
davar55 May 2004 New York City 5·7·11² Posts	Just starting to read your refreshing work. Two paragraphs in and it looks and reads clean.

2015-11-29, 05:06	#7
Dubslow Basketry That Evening! "Bunslow the Bold" Jun 2011 40<A<43 -89<O<-88 3·29·83 Posts	You still see missing parens? Where? The a->b typo's been fixed

2015-11-29, 22:44	#11
henryzz Just call me Henry "David" Sep 2007 Liverpool (GMT/BST) 37×163 Posts	I am slowly working through your page. It looks very good so far. Thought I would point out another form of guide: Take $2^8773$ for example $\sigma(73)=74=237$ . This means that $37 \| a(2^877337n)$ . This process can be repeated to get $2^8773371953n$ 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 $2^335n$ mutate directly to the form $2^33^25n$ ? It rings a bell that there are restrictions on how 2^335*n can mutate. Forgive me if this is mentioned. I have only properly read up to section 3 and glanced at the rest.