20160917, 05:29  #1 
May 2004
2^{2}·79 Posts 
Carmichael numbers
561 may be a Carmichael number in the ring of integers; but it is only pseudoprime in the ring of Gaussian integers!

20160917, 08:57  #2 
Dec 2012
The Netherlands
2^{2}×5×83 Posts 

20161007, 04:03  #3 
May 2004
2^{2}×79 Posts 
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. 
20161007, 19:16  #4  
Aug 2006
3^{2}×5×7×19 Posts 
Quote:
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. 

20161020, 05:42  #5 
May 2004
2^{2}·79 Posts 
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. 
20161020, 08:48  #6  
Sep 2003
2^{2}·647 Posts 
Quote:
561 = 3 Γ 187 3 is the first Mersenne prime (2^{2} β 1) 3 is also a Mersenne prime exponent, if that's what you meant (2^{3} β 1 = 7) Last fiddled with by GP2 on 20161020 at 08:48 

20161020, 10:38  #7 
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
10100110010110_{2} Posts 

20161020, 10:45  #8 
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
2·5,323 Posts 
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. 
20161020, 11:02  #9 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
1C35_{16} Posts 
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. 
20170213, 10:35  #10 
May 2004
2^{2}×79 Posts 

20170921, 04:53  #11 
May 2004
2^{2}·79 Posts 
Carmichal numbers
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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