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Old 2008-12-03, 16:02   #4
R. Gerbicz
 
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"Robert Gerbicz"
Oct 2005
Hungary

30518 Posts
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Quote:
Originally Posted by devarajkandadai View Post
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
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