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 2016-09-17, 05:29 #1 devarajkandadai     May 2004 22·79 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!
2016-09-17, 08:57   #2
Nick

Dec 2012
The Netherlands

3·601 Posts

Quote:
 Originally Posted by devarajkandadai 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?

 2016-10-07, 04:03 #3 devarajkandadai     May 2004 31610 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.
2016-10-07, 19:16   #4
CRGreathouse

Aug 2006

598710 Posts

Quote:
 Originally Posted by devarajkandadai 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.

 2016-10-20, 05:42 #5 devarajkandadai     May 2004 22×79 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.
2016-10-20, 08:48   #6
GP2

Sep 2003

A1D16 Posts

Quote:
 Originally Posted by devarajkandadai 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 ร 187

3 is the first Mersenne prime (22 โ 1)

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

Last fiddled with by GP2 on 2016-10-20 at 08:48

2016-10-20, 10:38   #7
xilman
Bamboozled!

"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across

266108 Posts

Quote:
 Originally Posted by GP2 The first Carmichael number is 561. 561 = 3 ร 187 3 is the first Mersenne prime (22 โ 1) 3 is also a Mersenne prime exponent, if that's what you meant (23 โ 1 = 7)
Did he mean not all prime factors?

2016-10-20, 10:45   #8
xilman
Bamboozled!

"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across

23×31×47 Posts

Quote:
 Originally Posted by xilman 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.

 2016-10-20, 11:02 #9 Dubslow Basketry That Evening!     "Bunslow the Bold" Jun 2011 40
2017-02-13, 10:35   #10

May 2004

22·79 Posts

Quote:
 Originally Posted by GP2 The first Carmichael number is 561. 561 = 3 ร 187 3 is the first Mersenne prime (22 โ 1) 3 is also a Mersenne prime exponent, if that's what you meant (23 โ 1 = 7)
No- I had meant that all the prime factors of a Carmichael number cannot be Mersenne primes.

2017-09-21, 04:53   #11

May 2004

22·79 Posts
Carmichal numbers

Quote:
 Originally Posted by devarajkandadai 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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