20120526, 20:48  #12  
Noodles
"Mr. Tuch"
Dec 2007
Chennai, India
3·419 Posts 
Quote:
It states that if the number is odd, then multiply by 3, and then add 1, if it is even, then divide by 2. If this process is being continued again and again, then taking any starting number will eventually end in 1 always. I think that it is not being very tough in order to prove this fact, There are more being chances for this to be true, in turn that's what I will feel it repeatedly, in fact Quote:
ஏதாவது எண்ணை எடுத்துக்கொண்டு தொடங்குங்கள். அதை ஒற்றைப்படையாக இருந்தால், மூன்றால் பெருக்கிவிட்டு, ஒன்றை சேருங்கள். இரட்டைப்படையாக இருந்தால், இரண்டால் வகுங்கள். இந்த செயலை தொடர்ந்தால், எல்லாம் எண்களும் எப்பொழுதும் இறுதியில் ஒன்றில் வந்து முடிவடையும் என்று இந்த ஊகம் குறிக்கிறது. நான் இதை உண்மையாக இருக்க அதிக வாய்ப்புகள் இருக்கிறது என்று நினைக்கிறேன். அது போல், இதை நிரூபிக்கவும் மிகவும் எளிதாக இருக்கும் என்று எனக்கு தோன்றுகிறது. சரி? Last fiddled with by Raman on 20120526 at 21:12 

20120526, 21:16  #13  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Quote:


20120526, 21:23  #14  
"Garambois JeanLuc"
Oct 2011
France
10001000001_{2} Posts 
Quote:
You are wrong ! It's a problem because the demonstration is in french. And the translation on the forum is wrong. If q divides n and q don't divide n^2 then : If n=qA and q and A are coprime Then : sigma(n)=(1+q) sigma(A) And because q==1[m] then m will divide (q+1) So m will divide sigma(n) You have to take q==1[m], it is written in the french demonstration. Did you read the french demonstration or the english demonstration on the forum before posting this message ??? JeanLuc 

20120526, 21:46  #15 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
I'm working on making a (hopefully half decent) translation of the first PDF. I don't understand the first sentence of the proof of theorem 1. "Soient k 2 e/qui vaut m Ak". What's the 'e/qui'? Does it mean something like "Let k>=2 _along with_ an m \in A_k"?
Another note: In the first paragraph, it defines A_k for all l >= 1, but I don't see an l in the definition of the set. (Also, I think there's a typo 'pout' > 'pour' in there.) Also, it's slow going because half the time is spent reproducing the tex. 
20120526, 22:21  #16 
"Robert Gerbicz"
Oct 2005
Hungary
5×17×19 Posts 

20120526, 23:19  #17 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Here's a stab at translating the first PDF
http://www.aliquotes.com/Aliquote.pdf
I can't offer any commentary at all on the mathematics of it; I'm just a messenger (I still don't really understand the first sentence of the proof of Theorem 1.) I'd be willing to translate the second PDF, but it might go faster if I didn't have to reproduce all the TeX... (hint hint ) Edit: It's much fancier. This might take a while, but it'll be an interesting challenge. Can I get access to the original *TeX please? My last note is to be careful about super and subscripts; this forum's TeXrenderer can't handle multiple nested levels of supers/subs very well. I'd recommend reading this simultaneously with the original PDF  the su*s are clearer there.  Let be the kth prime number. Let and for all , let where is the Euler function. For all , let be the set of integers whose factorization starts with the "driver" . Let . Theorem 1 (Lenstra 76): For all , if , then . Proof: Let with . Then there exists a prime with such that . We note that . We show that for all , is divisible by . We have which in turn equals by construction of . Recall Fermat's little theorem: for all and a such that we have . We apply the theorem with and and we obtain , which is equivalent to . We directly deduce that . Thus . Corollary 1 (Garambois' 2nd conjecture): For all and all , there exists an Aliquot sequence such that for consecutive iterations, . Proof: We use Mertens' formula . In particular, diverges and the product goes to infinity. Therefore there exists an such that . We choose an element of . By Lensta's theorem, the first iterations of the Aliquot sequence starting at are respectively in . Let ; then is in and thus can be written as for an integer coprime to . We have and which is larger than by our choice of . Finally, is the sum of the divisors of , here including , so . Thus , whence we conclude . Last fiddled with by Dubslow on 20120526 at 23:33 
20120526, 23:45  #18  
"Robert Gerbicz"
Oct 2005
Hungary
5×17×19 Posts 
Quote:
The proof is good, but if you know Lenstra's theorem then this proof is really triviality. 

20120527, 05:36  #19 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
For those who don't know an ounce of French...
Title: "On the density of integers n divisible by a certain integer m such that m does not divide ."
1 Introduction "The purpose of this paper is to present the proof of a theorem concerning the density of integers n divisible by a certain integer m such that m doesn't divide . We will demonstrate that this density is null by giving the asymptotic density of these numbers n less than a real x." Translator's note: Hmm.. there's some way to translate this that I'm missing. Asymptotic complexity? 2 Notations and Definitions "In all of the following: Let be a fixed natural integer 3. and will designate positive real numbers. We define sumofdivisors and sumofproperdivisors functions like so: And and we also define the function : Euler's totient function which counts the number of integers that are coprime to ." TN: There's a typo: "et on défnit" > "et on définit" 3 Theorem 1 "The (asymptotic) density of the integers n divisible by m such that m doesn't divide ( ) is null (0). Further, we give the following [asymptotic density][asymptotic complexity] for any sufficiently large real number x: . TN: Sorry about the dagger, I couldn't find a "does not divide" symbol that the forum would render. TN2: I guess I'll slowly translate the thing over the course of a few days or something, then copy and paste the tex into one post. Last fiddled with by Dubslow on 20120527 at 05:55 Reason: adding part 2 
20120527, 05:53  #20 
"Vincent"
Apr 2010
Over the rainbow
2^{2}×7×103 Posts 
As I said earlier, I shouldn't try to translate mathinvolving thing.

20120527, 05:55  #21  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Quote:
Quote:
I suppose that's why he called it a corollary Last fiddled with by Dubslow on 20120527 at 05:56 

20120527, 10:40  #22  
"Garambois JeanLuc"
Oct 2011
France
10001000001_{2} Posts 
Quote:
Wrong (old post) : If q divides n and q don't divide n^2 then : Right : If q divides n and q^2 don't divide n then : JeanLuc 

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