Thread: 3-factor Carmichael numbers View Single Post 2008-12-03, 16:02   #4
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

30518 Posts Quote:
 Originally Posted by devarajkandadai NECESSARY & SUFFICIENT CONDITIONS FOR A THREE-FACTOR COMPOSITE NUMBER WITH FOLLOWING SHAPE TO BE A CARMICHAEL NUMBER Let N, the composite number, have the shape (2m+1)(10m+1)(16m+1). Here m belongs to N. The necessary and sufficient conditions: a) (80m^2 + 53m + 7)/20 should be an integer b) The values of m which render the above an integer should also render 2m + 1, 10m + 1 and 16m + 1 prime. This is a corollary of the Devaraj-Pomerance-Maxal theorem (ref:: www.crorepatibaniye.com/failurefunctions
This is trivial, because lcm(p1-1,p2-1,p3-1)=80m. n is Carmichael number if and only if n-1 is divisible by 80m, n is squarefree and odd number (this is the Korselt theorem), (n-1)/(80m)=(80*m^2 + 53*m + 7)/20

Why you don't write: (80m^2 + 53m + 7)%20==13m+7==13*(m-1)==0 mod 20 so the simple condition is that m-1 is divisble by 20.

Last fiddled with by R. Gerbicz on 2008-12-03 at 16:03  