20120526, 14:12  #1 
Oct 2011
367 Posts 
A new theorem about aliquot sequences
Hello everybody,
Since one year, you know that : What was presented on the website www.aliquotes.com as the second conjecture of Garambois has become BarbulescuGarambois' theorem i.e.: There is an increasing aliquot sequence at each iteration of a factor at least k during i successive iterations, with k and i having any magnitude whatsoever. The demonstration by Razvan Barbulescu of the theorem can be found by clicking on the link. But since may 15th 2012, there is a second theorem which will probably be important about guides and drivers in aliquot sequences : What was presented on the website www.aliquotes.com as the third conjecture of Garambois has become ChtaibiGarambois' theorem i.e.: A guide (or a driver) in an aliquot sequence is all the more likely to be preserved with iterations going along as the terms of the aliquot sequence are getting bigger. The demonstration by Youssef Chtaibi of the theorem can be found by clicking on the link. Razvan Barbulescu verified this demonstration. I'm sorry, but those two demonstrations are in french ! JeanLuc Garambois 
20120526, 14:46  #2 
Apr 2010
Over the rainbow
2468_{10} Posts 
hmm si je puis me permettre, une petite faute d'orthographe
Apres (6) juste au dessus de (11) Maintenant on va considerer le poduit suivant : le produit, surement? En ce qui concerne les maths, j'ai pas la formation pour tout comprendre. je laisse ça aux autres. 
20120526, 15:15  #3  
Oct 2011
367 Posts 
Quote:
On m'a aussi signalé quelques fautes d'accents auxquelles je n'avais pas fait attention. Je contacte M. Chtaibi qui lui seul a le fichier source pour qu'il rectifie. JeanLuc 

20120526, 15:34  #4 
Noodles
"Mr. Tuch"
Dec 2007
Chennai, India
3×419 Posts 
கொல்லாட்ஸ் ஊகம் நிரூபிக்கப்பட்டுவிட்டதா? இந்த அனுமானம் உண்மையா? எந்த தொடங்குகின்ற எண்ணும் திரும்ப திரும்ப மூன்று பெருக்கப்பட்டு ஒன்று சேர்க்கப்படும் போது ஒன்றில் முடிவடையும் என்று கூறுகிறது. அது உண்மையாக இருக்க அதிக வாய்ப்பு இருக்கிறது என்று எனக்கு தோன்றுகிறது.
Last fiddled with by Raman on 20120526 at 15:45 
20120526, 15:57  #5 
Apr 2010
Over the rainbow
2^{2}×617 Posts 
Sorry Raman, you'll have to write it in french or english... What i said was about a spelling mistake.
google translate activate!(poduit : produit, as product) Code:
kollats poduit நிரூபிக்கப்பட்டுவிட்டதா speculation? This assumption true? Counting begins when adding back any one of three multiplied together, which suggests that in the end. I feel that it is more likely to be true. Last fiddled with by firejuggler on 20120526 at 16:02 
20120526, 16:03  #6 
"Robert Gerbicz"
Oct 2005
Hungary
2^{3}×3×59 Posts 
Érdekes bizonyítás, szerintem be lehetne küldeni egy matematikai folyóiratnak is. Most már tudom, hogy mondják franciául a prímszámot. 3dik sorban vajon mit jelöl "val"al? Ezt nem definiálta a szerző. Lenstra bizonyítása nekem is beugrott, ő kevesebbet bizonyított. Magyarul is válaszolhattok.

20120526, 16:05  #7  
Noodles
"Mr. Tuch"
Dec 2007
Chennai, India
3×419 Posts 
Quote:
Thus, it translates into "Has the Collatz Conjecture been proved? Is it being true? It states that any starting number when repeatedly multiplied by three, and then added one, terminates in one. It appears to me that it is being very likely for this(it) to be true." For example, for some starting number N, repeatedly iterating the 3x+1 function one can try out to prove off something thereby establishing an upper bound result that the intermediate number value will not (cannot) go beyond N.2^{k} at all. Thus, hopefully that now you will be able to read and then understand it right now, So that you can reply to me quite clearly for that sentence itself, rather Last fiddled with by Raman on 20120526 at 16:48 

20120526, 16:14  #8 
Apr 2010
Over the rainbow
2^{2}·617 Posts 
ok, ok.. i won't write in french anymore.
Code:
hmm si je puis me permettre, une petite faute d'orthographe Apres (6) juste au dessus de (11) Maintenant on va considerer le poduit suivant : le produit, surement? En ce qui concerne les maths, j'ai pas la formation pour tout comprendre. je laisse ça aux autres. after (6) and just before (11) "Maintenant on va considerer le poduit suivant :" produit, for sure?. On the topic of math, i don't have the formation to understand all. I'll leave it to others. Last fiddled with by firejuggler on 20120526 at 16:15 
20120526, 17:15  #9  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·2,399 Posts 
Quote:


20120526, 18:18  #10 
Apr 2010
Over the rainbow
2^{2}·617 Posts 
Ok, I'm going to translate it (the second paper). Be warned, it will be rought.
Code:
Around the 'n' integer density dividable by a 'm' integer such as m doesn't divide sigma(n)(n) I Introduction The goal of this paper is to expose the proof about the 'n' integer density dividable by a 'm' integer such as m doesn't divide sigma(n)(n) that one will prove in the remainder of the paper that it is zero, giving an increase asymptotic of these numbers "n" wich are lower than a real number "x". (above is a barely modified version of google translation) 2 Notations et DefinitionsFor all that follow "m" is a natural integer >=3 "x" and "t" are positive real numbers We define the functions sum of divisors and sum of proper divisors as follows: sigma(n)= Sum(d) over d/k sigma'(n)=sigma(n)n and we also defines the function Phi (n): Euler indicator that counts the number integers n that are coprime to n. 3 Theorem 1 The density (asymptotic) of the integers n dividable by m such that m does not divide sigma'(n) is 0 (and Here start the problems) In addition to that, there is the following asymptotic majorationfor all large enough real "x" (cardinal equation wich I do not follow) 4 theoreme proof To demonstrate the theoreme, we have to prove the following lemme : 5 Lemme For all large enough real number x, we have : (not following either ) 6lemme proof let be x and t 2 real number large enough such as 1<<t<<x (strictly supperior?) It is clear that if a prime number q such as ...(no really) let be q1 < q2 < ... qi the prime nuùber such as qi mod m=1 . And from a Dirichlet's corrollary on prime number in the arythmetic progression... (more math...*drop tear*) No really I should stop. not being able to translate at all wouldn't be any better Last fiddled with by firejuggler on 20120526 at 18:37 
20120526, 18:57  #11 
"Robert Gerbicz"
Oct 2005
Hungary
2^{3}×3×59 Posts 
There are serious problems with the proof of lemma 5 (2nd article) even for cases where m is prime. As I can see you are trying to determine the complementer event: if q==1 mod m and qn and q^2 doesn't divide n then msigma(n).
But there are cases you left out in the counting: let m=5 and 2^3n and 2^4 doesn't divide n you can get that sigma(2^3)=15 divides sigma(n), so 5 also divides sigma(n) but 2==1 mod 5 is not true. For composites m there are much more problems with the proof. Last fiddled with by R. Gerbicz on 20120526 at 18:59 
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