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Old 2018-12-01, 06:44   #8
devarajkandadai's Avatar
May 2004

4748 Posts

Originally Posted by devarajkandadai View Post
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_1-1)*(N-1)/(p_2-1)......... (p_r-1) 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)
C) let N = (2*m+1)*(10*m+1)*(16*m+1)- here m is a natural nnumber. Then N is a Carmichael number if a) for a given value of m, 2*m+1, 10*m+1 and 16*m+1 are prime and b) 80*m^2 + 53*m + 7 is exactly divisible by 20.
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