mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Miscellaneous Math

Reply
 
Thread Tools
Old 2005-02-14, 09:28   #1
devarajkandadai
 
devarajkandadai's Avatar
 
May 2004

4748 Posts
Default Programming a Conjecture

At the outset I must confess that my knowledge of computer programming
is nil.
Q: Is it possible to programme my second conjecture on site:

www.crorepathibaniye.com/failurefunctions

in order to generate Carmichael Numbers?

A.K. Devaraj
devarajkandadai is offline   Reply With Quote
Old 2005-02-14, 18:08   #2
Peter Nelson
 
Peter Nelson's Avatar
 
Oct 2004

232 Posts
Default Can't access your conjecture

Quote:
Originally Posted by devarajkandadai
At the outset I must confess that my knowledge of computer programming
is nil.
Q: Is it possible to programme my second conjecture on site:

www.crorepathibaniye.com/failurefunctions

in order to generate Carmichael Numbers?

A.K. Devaraj
We would like to take a look at your work if we could find it.

Have tried the link you posted and pinging the hostname with no response. Have also tried spelling variations.

Please could you check the spelling you typed and revise if needed.

Perhaps your server is temporarily unavailable or needs restarting.

Alternatively, post an overview of your work in the forum.

Once we can view your work we can advise on programming.

Regards, Peter
Peter Nelson is offline   Reply With Quote
Old 2005-02-15, 11:02   #3
maxal
 
maxal's Avatar
 
Feb 2005

3748 Posts
Default

Quote:
Originally Posted by devarajkandadai
At the outset I must confess that my knowledge of computer programming
is nil.
Q: Is it possible to programme my second conjecture on site:

www.crorepathibaniye.com/failurefunctions

in order to generate Carmichael Numbers?

A.K. Devaraj
First off, you meant http://www.crorepatibaniye.com/failu...onjecture2.asp, didn't you?

Second, your conjecture can be reformulated as
Quote:
Let N=p1*...*pr be a product of r prime factors. Then N is Carmichael number iff gcd(p1-1,p2-1,...,pr-1)^2*(N-1)^(r-2) is divisable by phi(N)=(p1-1)*...*(pr-1).
Third, I've tested your conjecture for all numbers less than 10^7 using a PARI program listed below.

It happens that the conjecture works in one direction, namely, for all Carmichael numbers N=p1*...*pr less than 10^7, gcd(p1-1,p2-1,...,pr-1)^2*(N-1)^(r-2) is divisable by phi(N)=(p1-1)*...*(pr-1).
The opposite is not true in general. There are counterexamples like
11305 = 5*7*17*19
39865 = 5*7*17*67
96985 = 5*7*17*163
401401 = 7*11*13*401
which satisfy the divisibility condition not being Carmichael numbers.

Code:
{ test() = 
for(n=2,10^8, 
  f=factorint(n); 
  if(vecmax(f[,2])>1,next); 
  f=f[,1]; r=length(f); 
  realCM=1; 
  d=n-1; 
  for(i=1,r, d=gcd(d,f[i]-1); if((n-1)%(f[i]-1),realCM=0)); 
  if( (((n-1)^(r-2)*d^2)%eulerphi(n)==0) != realCM, print(n," ",realCM," ",f)) 
) }

Last fiddled with by maxal on 2005-02-15 at 11:06
maxal is offline   Reply With Quote
Old 2005-02-15, 13:49   #4
devarajkandadai
 
devarajkandadai's Avatar
 
May 2004

1001111002 Posts
Default Programming a conjecture

Thank u very much, Maxal.After studying your reply I may have further questions;is it o,k,?
A.K. Devaraj
devarajkandadai is offline   Reply With Quote
Old 2005-02-16, 04:03   #5
maxal
 
maxal's Avatar
 
Feb 2005

3748 Posts
Default

Quote:
Originally Posted by devarajkandadai
Thank u very much, Maxal.After studying your reply I may have further questions;is it o,k,?
A.K. Devaraj
Sure. Just post your questions here, or send me a PM if you like.
maxal is offline   Reply With Quote
Old 2005-02-16, 05:10   #6
devarajkandadai
 
devarajkandadai's Avatar
 
May 2004

22·79 Posts
Default

Quote:
Originally Posted by devarajkandadai
At the outset I must confess that my knowledge of computer programming
is nil.
Q: Is it possible to programme my second conjecture on site:

www.crorepathibaniye.com/failurefunctions

in order to generate Carmichael Numbers?

A.K. Devaraj
Dear Maxal,
I am afraid you have not studied the conjecture; it is not only one of the divisibility tests that must be satisfied BUT ALL OF THEM in order to fulfil
the "necessary & sufficient" conditions.Kindly try again and you will find that
11305 FAILS one of these tests and hence can be rejected from the
list of C.N.S.Regards
A.K. Devaraj
devarajkandadai is offline   Reply With Quote
Old 2005-02-16, 06:37   #7
maxal
 
maxal's Avatar
 
Feb 2005

22×32×7 Posts
Default

Quote:
Originally Posted by devarajkandadai
I am afraid you have not studied the conjecture; it is not only one of the divisibility tests that must be satisfied BUT ALL OF THEM in order to fulfil
the "necessary & sufficient" conditions.Kindly try again and you will find that
11305 FAILS one of these tests and hence can be rejected from the list of C.N.S.
It does fulfil *ALL* the conditions. Look:
Code:
(5-1)*(11305-1)^2 / ((7-1)*(17-1)*(19-1)) = 295788
(7-1)*(11305-1)^2 / ((5-1)*(17-1)*(19-1)) = 665523
(17-1)*(11305-1)^2 / ((5-1)*(7-1)*(19-1)) = 4732608
(19-1)*(11305-1)^2 / ((5-1)*(7-1)*(17-1)) = 5989707
So all quotients are integer.

And as I stated before, there is a simpler equivalent formulation of your conjecture. Of course, 11305 satisfy its condition as well:
gcd(5-1,7-1,17-1,19-1) = 2
and 2^2*(11305-1)^2 / ((5-1)*(7-1)*(17-1)*(19-1)) = 73947, an integer number.

Last fiddled with by maxal on 2005-02-16 at 06:39
maxal is offline   Reply With Quote
Old 2005-02-16, 13:35   #8
devarajkandadai
 
devarajkandadai's Avatar
 
May 2004

1001111002 Posts
Default programming a conjecture

Dear Maxal,
Yes I did recheck & found you are correct even before u replied to my post.
Is there a site where we can obtain all the 4-factor Car.Numbrs?
Thanking you,
A.K. Devaraj
devarajkandadai is offline   Reply With Quote
Old 2005-02-17, 07:47   #9
maxal
 
maxal's Avatar
 
Feb 2005

22·32·7 Posts
Default

Quote:
Originally Posted by devarajkandadai
Is there a site where we can obtain all the 4-factor Car.Numbrs?
Try this: http://www.research.att.com/projects/OEIS?Anum=A074379
I can generate more if needed.
maxal is offline   Reply With Quote
Old 2005-02-25, 20:24   #10
maxal
 
maxal's Avatar
 
Feb 2005

FC16 Posts
Default new sequences

Dear devarajkandadai,

I've added two sequences related to your conjecture to OEIS:

http://www.research.att.com/projects/OEIS?Anum=A104016

http://www.research.att.com/projects/OEIS?Anum=A104017
maxal is offline   Reply With Quote
Old 2005-03-09, 02:50   #11
devarajkandadai
 
devarajkandadai's Avatar
 
May 2004

22×79 Posts
Default

Quote:
Originally Posted by maxal
Dear devarajkandadai,

I've added two sequences related to your conjecture to OEIS:

http://www.research.att.com/projects/OEIS?Anum=A104016

http://www.research.att.com/projects/OEIS?Anum=A104017
Dear Maxal,
First of all I must thank you for giving my name to the set of numbers generated by my conjecture.Secondly I must thank you for the sequences themselves.
Regards
A.K. Devaraj
devarajkandadai is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Programming questions Xyzzy Programming 76 2015-01-24 04:35
Need some help on php programming pinhodecarlos Programming 2 2012-07-23 18:17
New to programming. What to do? lorgix Miscellaneous Math 9 2010-12-08 22:22
plz, help me in c programming alaa Homework Help 12 2007-06-12 22:17
Programming a Theorem devarajkandadai Miscellaneous Math 11 2005-03-30 05:08

All times are UTC. The time now is 22:42.

Thu Oct 29 22:42:49 UTC 2020 up 49 days, 19:53, 1 user, load averages: 2.54, 2.10, 2.09

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.