Quote:
Originally Posted by devarajkandadai
NECESSARY & SUFFICIENT CONDITIONS FOR A THREEFACTOR 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 DevarajPomeranceMaxal theorem (ref::
www.crorepatibaniye.com/failurefunctions

This is trivial, because lcm(p11,p21,p31)=80m. n is Carmichael number if and only if n1 is divisible by 80m, n is squarefree and odd number (this is the Korselt theorem), (n1)/(80m)=(80*m^2 + 53*m + 7)/20
Why you don't write: (80m^2 + 53m + 7)%20==13m+7==13*(m1)==0 mod 20 so the simple condition is that m1 is divisble by 20.