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:

[url]www.crorepathibaniye.com/failurefunctions[/url]

in order to generate Carmichael Numbers?

A.K. Devaraj

Can't access your conjecture
 

[QUOTE=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:

[url]www.crorepathibaniye.com/failurefunctions[/url]

in order to generate Carmichael Numbers?

A.K. Devaraj[/QUOTE]

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

[QUOTE=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:

[url]www.crorepathibaniye.com/failurefunctions[/url]

in order to generate Carmichael Numbers?

A.K. Devaraj[/QUOTE]

First off, you meant [url]http://www.crorepatibaniye.com/failurefunctions/conjecture2.asp[/url], 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).[/quote]

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))
) }[/CODE]

Programming a conjecture
 

Thank u very much, Maxal.After studying your reply I may have further questions;is it o,k,?
A.K. Devaraj

[QUOTE=devarajkandadai]Thank u very much, Maxal.After studying your reply I may have further questions;is it o,k,?
A.K. Devaraj[/QUOTE]
Sure. Just post your questions here, or send me a PM if you like.

[QUOTE=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:

[url]www.crorepathibaniye.com/failurefunctions[/url]

in order to generate Carmichael Numbers?

A.K. Devaraj[/QUOTE]
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

[QUOTE=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.[/QUOTE]
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[/code]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.

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

[QUOTE=devarajkandadai]Is there a site where we can obtain all the 4-factor Car.Numbrs?
[/QUOTE]
Try this: [url]http://www.research.att.com/projects/OEIS?Anum=A074379[/url]
I can generate more if needed.

new sequences
 

Dear devarajkandadai,

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

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

[url]http://www.research.att.com/projects/OEIS?Anum=A104017[/url]

[QUOTE=maxal]Dear devarajkandadai,

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

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

[url]http://www.research.att.com/projects/OEIS?Anum=A104017[/url][/QUOTE]
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