A wrong track to demonstrate that the Catalan's conjecture is false

garambois · 2015-10-20, 12:41

Hello everyone,

There is a new idea to show that the Catalan's aliquot sequences conjecture is false.

Proposition 1 :
The Catalan's aliquot sequences conjecture is false : there are aliquot sequences that grow indefinitely.

Proposition 2 :
Let p₀ to be a prime number and k to be an integer with p₀>k>2.
One iterates as follows : p_i+1=(k-1)•p_i+k.
There are a number k, and a prime number p₀ such that p_i is prime for all i.

If the proposition 2 is true, then the proposition 1 is true and Catalan's aliquot sequences conjecture is false.

To see the proof, but sorry, the article is in french, click here :

http://www.aliquotes.com/infirmer_catalan_2.pdf

Jean-Luc Garambois

LaurV · 2015-10-20, 14:25

Of course Proposition 2 implies Proposition 1, but Proposition 2 seems a bit too strong for me.
Subjectively, I assume it is false, because I "believe" the Catalan conjecture to be true

If that is true, and we find a k and p, wouldn't allow us to generate an infinite number of increasing primes?

fivemack · 2015-10-20, 14:29

You have demonstrated that a statement sufficiently strong that the normal probabilistic-number-theory arguments would say it was almost certainly false (indeed, isn't there a guarantee just by looking mod k-1 that you can't get to length more than k), implies Catalan's conjecture.

(p0=29 k=21 lasts for four steps; p0=103 k=10 lasts seven; p0=33851 k=1050 seems to be the smallest p0 that lasts nine)

Anonuser · 2015-10-20, 15:22

Quote:

Originally Posted by garambois

Hello everyone,

There is a new idea to show that the Catalan's aliquot sequences conjecture is false.

Proposition 1 :
The Catalan's aliquot sequences conjecture is false : there are aliquot sequences that grow indefinitely.

Proposition 2 :
Let p₀ to be a prime number and k to be an integer with p₀>k>2.
One iterates as follows : p_i+1=(k-1)•p_i+k.
There are a number k, and a prime number p₀ such that p_i is prime for all i.

If the proposition 2 is true, then the proposition 1 is true and Catalan's aliquot sequences conjecture is false.

To see the proof, but sorry, the article is in french, click here :

http://www.aliquotes.com/infirmer_catalan_2.pdf

Jean-Luc Garambois

I am afraid that Proposition 2 is not true (see the (corrected) attachment).

garambois · 2015-10-20, 15:23

Quote:

Originally Posted by LaurV

Of course Proposition 2 implies Proposition 1, but Proposition 2 seems a bit too strong for me.
Subjectively, I assume it is false, because I "believe" the Catalan conjecture to be true

If that is true, and we find a k and p, wouldn't allow us to generate an infinite number of increasing primes?

If proposition 2 is false, that would not implies that Catalan's conjecture is true, but OK, if proposition 2 is false, perhaps Catalan's conjecture is true !

And I don't know if it is forbidden by number theory to generate an infinite number of increasing primes by this iteration process (proposition 2).

fivemack · 2015-10-20, 15:52

Quote:

Originally Posted by garambois

If proposition 2 is false, that would not implies that Catalan's conjecture is true, but OK, if proposition 2 is false, perhaps Catalan's conjecture is true !

And I don't know if it is forbidden by number theory to generate an infinite number of increasing primes by this iteration process (proposition 2).

Yes, it's forbidden - look at the remainders of the elements in your series modulo k-1, they go up by 1 at each iteration so eventually one will be divisible by k-1 !

science_man_88 · 2015-10-20, 15:53

Quote:

Originally Posted by garambois

If proposition 2 is false, that would not implies that Catalan's conjecture is true, but OK, if proposition 2 is false, perhaps Catalan's conjecture is true !

And I don't know if it is forbidden by number theory to generate an infinite number of increasing primes by this iteration process (proposition 2).

Wikipedia suggest that iterative functions produce:

${(k-1)}^{i+1}p_i+\frac{(k-1)^{i+1}-1}{(k-2)}k$ at iteration i+1

garambois · 2015-10-20, 15:59

OK, MY PROPOSITION 2 IS FALSE !

Sorry.
Thank you for your help.
Like this, we will not continue the research in this wrong direction.

Is it possible to write near the title "PROPOSITION RETIRED" ?

I received a mail from Paul Zimmermann too because my proposition 2 is false.

Jean-Luc Garambois

garambois · 2015-10-20, 16:37

Thank you for the new title !
Jean-Luc

LaurV · 2015-10-21, 01:32

At most what you can do, you can demonstrate that the string can be made as long as you want (i.e. given any integer n, there is a k and p0 such as the string contains at least n prime terms). This will result in the fact that there are aliquot sequences which can be made arbitrary high (or as long or as high as we want). But this is demonstrated already, long ago.

garambois · 2015-10-21, 08:22

Now, I will explore other ways to try to disprove Catalan's conjecture.
Because, I really think it is false.
I have other ideas...
But I will be more careful in the future when I will propose a new conjecture. This will serve me a lesson.
On the other hand, here, I had a very quick response that showed me that I was mistaken.
This saved me from losing too much time on this issue that some of you knew the answer !
Again thank you for your answers.

Jean-Luc

2015-10-20, 12:41	#1
garambois "Garambois Jean-Luc" Oct 2011 France 2⁷×5 Posts	A wrong track to demonstrate that the Catalan's conjecture is false Hello everyone, There is a new idea to show that the Catalan's aliquot sequences conjecture is false. Proposition 1 : The Catalan's aliquot sequences conjecture is false : there are aliquot sequences that grow indefinitely. Proposition 2 : Let p₀ to be a prime number and k to be an integer with p₀>k>2. One iterates as follows : p_i+1=(k-1)•p_i+k. There are a number k, and a prime number p₀ such that p_i is prime for all i. If the proposition 2 is true, then the proposition 1 is true and Catalan's aliquot sequences conjecture is false. To see the proof, but sorry, the article is in french, click here : http://www.aliquotes.com/infirmer_catalan_2.pdf Jean-Luc Garambois

2015-10-20, 14:25	#2
LaurV Romulan Interpreter Jun 2011 Thailand 7×1,373 Posts	Of course Proposition 2 implies Proposition 1, but Proposition 2 seems a bit too strong for me. Subjectively, I assume it is false, because I "believe" the Catalan conjecture to be true If that is true, and we find a k and p, wouldn't allow us to generate an infinite number of increasing primes? Last fiddled with by LaurV on 2015-10-20 at 14:27

2015-10-20, 15:59	#8
garambois "Garambois Jean-Luc" Oct 2011 France 2⁷·5 Posts	OK, MY PROPOSITION 2 IS FALSE ! Sorry. Thank you for your help. Like this, we will not continue the research in this wrong direction. Is it possible to write near the title "PROPOSITION RETIRED" ? I received a mail from Paul Zimmermann too because my proposition 2 is false. Jean-Luc Garambois Last fiddled with by garambois on 2015-10-20 at 16:03

2015-10-21, 08:22	#11
garambois "Garambois Jean-Luc" Oct 2011 France 280₁₆ Posts	Now, I will explore other ways to try to disprove Catalan's conjecture. Because, I really think it is false. I have other ideas... But I will be more careful in the future when I will propose a new conjecture. This will serve me a lesson. On the other hand, here, I had a very quick response that showed me that I was mistaken. This saved me from losing too much time on this issue that some of you knew the answer ! Again thank you for your answers. Jean-Luc Last fiddled with by garambois on 2015-10-21 at 08:22

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2015-10-20, 14:29	#3
fivemack (loop (#_fork)) Feb 2006 Cambridge, England 7²·131 Posts	You have demonstrated that a statement sufficiently strong that the normal probabilistic-number-theory arguments would say it was almost certainly false (indeed, isn't there a guarantee just by looking mod k-1 that you can't get to length more than k), implies Catalan's conjecture. (p0=29 k=21 lasts for four steps; p0=103 k=10 lasts seven; p0=33851 k=1050 seems to be the smallest p0 that lasts nine)

2015-10-20, 16:37	#9
garambois "Garambois Jean-Luc" Oct 2011 France 2⁷·5 Posts	Thank you for the new title ! Jean-Luc

2015-10-21, 01:32	#10
LaurV Romulan Interpreter Jun 2011 Thailand 22613₈ Posts	At most what you can do, you can demonstrate that the string can be made as long as you want (i.e. given any integer n, there is a k and p0 such as the string contains at least n prime terms). This will result in the fact that there are aliquot sequences which can be made arbitrary high (or as long or as high as we want). But this is demonstrated already, long ago.