20181030, 12:08  #1 
May 2004
474_{8} Posts 
Carmichael numbers and Šimerka numbers
As you are aware Carmichael numbers pertain to the property of composite numbers
behaving like prime numbers with regard to Fermat's theorem. They are Devaraj numbers I.e. if N = p_1*p_2....p_r ( where p_i is prime) then (P_11)*(N1)/(p_21)......... (p_r1) is an integer. See A104016 and A104017. a) conjecture: the least value of k, the degree to which atleast two of a Devaraj number's prime factors are Inverses, is 2 (example 561 = 3*11*17 here 3 and 17 are inverses (mod 5^2). b) 5 and 11 are impossible cofactors of Devaraj numbers (including Carmichael numbers). (to be continued) 
20181030, 14:18  #2  
May 2004
474_{8} Posts 
Carmichael numbers and Devaraj numbers
Quote:
Last fiddled with by devarajkandadai on 20181030 at 14:19 Reason: Corrected a slip 

20181030, 14:24  #3  
May 2004
13C_{16} Posts 
Carmichael numbers and Devaraj numbers
Quote:


20181102, 05:54  #4 
May 2004
2^{2}×79 Posts 
Carmichael numbers and Devaraj numbers

20181104, 05:30  #5 
May 2004
2^{2}×79 Posts 
Carmichael numbers and Devaraj numbers

20181105, 04:16  #6 
May 2004
2^{2}·79 Posts 

20181105, 06:05  #7 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
23·439 Posts 
Well, 5 and 7469128023...77_{<181>} are goddamn inverses of degree 600.
131 and 1289338297...07_{<1808>} are inverses of degree 6002. 3 and (4025*2^66666+1)/3 are inverses of degree 66666. 7 and (3*2^320008+1)/7 are inverses of degree 320008. There are thousands of similar anecdotal cases. Do you have a point to make other than torture random semiprime numbers? 
20181201, 06:44  #8  
May 2004
474_{8} Posts 
Quote:


20181201, 16:43  #9  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
23×439 Posts 
Quote:
With a difference that Chernick proved his and you are "just saying". To what limit did you even test it? 

20181201, 17:25  #10 
"Forget I exist"
Jul 2009
Dartmouth NS
2^{2}×7^{2}×43 Posts 
which only works if q is 1 mod 3, because the first defeats 0 mod 3 and the others fail for 2 mod 3.

20181201, 20:59  #11 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
10097_{10} Posts 

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