A new theorem about aliquot sequences

[garambois]

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 Barbulescu-Garambois' 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 Chtaibi-Garambois' 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 !

Jean-Luc Garambois

[firejuggler]

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.

[garambois]

Originally Posted by firejuggler

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.

Merci pour ces remarques.
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.
Jean-Luc

[Raman]

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.

[firejuggler]

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)

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.

[R. Gerbicz]

Érdekes bizonyítás, szerintem be lehetne küldeni egy matematikai folyóiratnak is. Most már tudom, hogy mondják franciául a prímszámot. 3-dik sorban vajon mit jelöl "val"-al? Ezt nem definiálta a szerző. Lenstra bizonyítása nekem is beugrott, ő kevesebbet bizonyított. Magyarul is válaszolhattok.

[Raman]

Originally Posted by firejuggler

Sorry Raman, you'll have to write it in french or english... What i said was about a spelling mistake.

You wrote in a language that I cannot read at all. So that I decided to write in my own native local language. Thus, were you able to read it? Did your PC rather render it properly, mainly. If you can obfuscate it, whatever you post, don't I know how to do it?

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

[firejuggler]

ok, ok.. i won't write in french anymore.

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.

If you allow me, a small spelling mistake,
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.

[Dubslow]

Originally Posted by firejuggler

En ce qui concerne les maths, j'ai pas la formation pour tout comprendre. je laisse ça aux autres.

Mais surement you can translate it? Pour les termes que vous (et moi) ne comprennez pas, les maths sont assez precise qu'on peut utiliser (par exemple) Wikipedia pour une traduction des termes individuelles. (Je l'essaierais moi-même, mais en ce moment je vais au cinéma avec ma famille; je suis sur mon télé portable )

[firejuggler]

Ok, I'm going to translate it (the second paper). Be warned, it will be rought.

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

[R. Gerbicz]

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 q|n and q^2 doesn't divide n then m|sigma(n).

But there are cases you left out in the counting: let m=5 and 2^3|n 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.