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 2005-02-14, 09:28 #1 devarajkandadai     May 2004 4748 Posts 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
2005-02-14, 18:08   #2
Peter Nelson

Oct 2004

232 Posts

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.

Regards, Peter

2005-02-15, 11:02   #3
maxal

Feb 2005

3748 Posts

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

 2005-02-15, 13:49 #4 devarajkandadai     May 2004 1001111002 Posts Programming a conjecture Thank u very much, Maxal.After studying your reply I may have further questions;is it o,k,? A.K. Devaraj
2005-02-16, 04:03   #5
maxal

Feb 2005

3748 Posts

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.

2005-02-16, 05:10   #6

May 2004

22·79 Posts

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

2005-02-16, 06:37   #7
maxal

Feb 2005

22×32×7 Posts

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

 2005-02-16, 13:35 #8 devarajkandadai     May 2004 1001111002 Posts 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
2005-02-17, 07:47   #9
maxal

Feb 2005

22·32·7 Posts

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.

 2005-02-25, 20:24 #10 maxal     Feb 2005 FC16 Posts 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
2005-03-09, 02:50   #11

May 2004

22×79 Posts

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

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