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Old 2016-09-17, 05:29   #1
devarajkandadai
 
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Default Carmichael numbers

561 may be a Carmichael number in the ring of integers; but it is only pseudoprime in the ring of Gaussian integers!
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Old 2016-09-17, 08:57   #2
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Quote:
Originally Posted by devarajkandadai View Post
561 may be a Carmichael number in the ring of integers; but it is only pseudoprime in the ring of Gaussian integers!
And in the ring of Eisenstein integers?
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Old 2016-10-07, 04:03   #3
devarajkandadai
 
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Default 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.
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Old 2016-10-07, 19:16   #4
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Quote:
Originally Posted by devarajkandadai View Post
A conjecture pertaining to CNs:
Go to Youtube and search for akdevaraj; prove or disprove a conjecture stated by me in my talk.
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.
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Old 2016-10-20, 05:42   #5
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Default 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.
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Old 2016-10-20, 08:48   #6
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Quote:
Originally Posted by devarajkandadai View Post
I will now state the conjecture: All the prime factors of a Carmichael number cannot be Mersenne primes.
The first Carmichael number is 561.

561 = 3 Γ— 187

3 is the first Mersenne prime (22 βˆ’ 1)

3 is also a Mersenne prime exponent, if that's what you meant (23 βˆ’ 1 = 7)

Last fiddled with by GP2 on 2016-10-20 at 08:48
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Old 2016-10-20, 10:38   #7
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Quote:
Originally Posted by GP2 View Post
The first Carmichael number is 561.

561 = 3 Γ— 187

3 is the first Mersenne prime (22 βˆ’ 1)

3 is also a Mersenne prime exponent, if that's what you meant (23 βˆ’ 1 = 7)
Did he mean not all prime factors?
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Old 2016-10-20, 10:45   #8
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Originally Posted by xilman View Post
Did he mean not all prime factors?
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.
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Old 2016-10-20, 11:02   #9
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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.
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Old 2017-02-13, 10:35   #10
devarajkandadai
 
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Quote:
Originally Posted by GP2 View Post
The first Carmichael number is 561.

561 = 3 Γ— 187

3 is the first Mersenne prime (22 βˆ’ 1)

3 is also a Mersenne prime exponent, if that's what you meant (23 βˆ’ 1 = 7)
No- I had meant that all the prime factors of a Carmichael number cannot be Mersenne primes.
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Old 2017-09-21, 04:53   #11
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Default Carmichal numbers

Quote:
Originally Posted by devarajkandadai View Post
561 may be a Carmichael number in the ring of integers; but it is only pseudoprime in the ring of Gaussian integers!
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.
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