Carmichael numbers
 

561 may be a Carmichael number in the ring of integers; but it is only pseudoprime in the ring of Gaussian integers!

[QUOTE=devarajkandadai;442816]561 may be a Carmichael number in the ring of integers; but it is only pseudoprime in the ring of Gaussian integers![/QUOTE]
And in the ring of Eisenstein integers?

Carmichael numbers
 

A conjecture pertaining to CNs:
Go to Youtube and search for akdevaraj; prove or disprove a conjecture stated by me in my talk.

[QUOTE=devarajkandadai;444436]A conjecture pertaining to CNs:
Go to Youtube and search for akdevaraj; prove or disprove a conjecture stated by me in my talk.[/QUOTE]

1. Find YouTube video.
2. Watch YouTube video, transcribe mathematical content.
3. Decipher the meaning of same.
4. Gather information: finite checking, literature search, heuristics.
5. Attempt to prove or disprove.

I'm willing to take a hack at #4 and #5 if others do #1 - #3.

Carmichael numbers -
 

I had suggested youtube in order to increase viewership of my video.
I will now state the conjecture: All the prime factors of a Carmichael number cannot be Mersenne primes.

[QUOTE=devarajkandadai;445413]I will now state the conjecture: All the prime factors of a Carmichael number cannot be Mersenne primes.[/QUOTE]

The first Carmichael number is 561.

561 = 3 × 187

3 is the first Mersenne prime (2[SUP]2[/SUP] − 1)

3 is also a Mersenne prime exponent, if that's what you meant (2[SUP]3[/SUP] − 1 = 7)

[QUOTE=GP2;445417]The first Carmichael number is 561.

561 = 3 × 187

3 is the first Mersenne prime (2[SUP]2[/SUP] − 1)

3 is also a Mersenne prime exponent, if that's what you meant (2[SUP]3[/SUP] − 1 = 7)[/QUOTE]

Did he mean not [B]all[/B] prime factors?

[QUOTE=xilman;445420]Did he mean not [B]all[/B] prime factors?[/QUOTE]The sentence "All the prime factors of a Carmichael number cannot be Mersenne primes." is ambigous.

It could be read (at least) as

For all Carmichael numbers C, the prime factors of C must include at least one prime which is not a Mersenne prime.

For all Carmichael numbers C, no prime factors of C may be a Mersenne prime.

There exists at least one Carmichael number C for which the set of prime factors of C does not include any Mersenne numbers.

The simplest interpretation is the middle one which GP2 provided a counter example for.

The first interpretation is a bit trickier to reach, requiring a more complex parsing of the grammar (and a bit of transposition is required to render this the simplest interpretation). It took me a few minutes to see how you could read it this way.

The third one is a bit of a stretch I think.

[QUOTE=GP2;445417]The first Carmichael number is 561.

561 = 3 × 187

3 is the first Mersenne prime (2[SUP]2[/SUP] − 1)

3 is also a Mersenne prime exponent, if that's what you meant (2[SUP]3[/SUP] − 1 = 7)[/QUOTE]

No- I had meant that all the prime factors of a Carmichael number cannot be Mersenne primes.

Carmichal numbers
 

[QUOTE=devarajkandadai;442816]561 may be a Carmichael number in the ring of integers; but it is only pseudoprime in the ring of Gaussian integers![/QUOTE]

Carmichael numbers are only pseudoprimes in the ring of Gaussian integers. However it is very easy to find appropriate bases for pseudoprimality. Let me illustrate only with an example. (3 + 187*I), (33+ 17*I), (51+11*I) and variations including conjugates are appropriate bases in the case of 561.