Quote:
Originally Posted by Dr Sardonicus

There appear to be two types of Higher order Carmichael numbers of order m:
n such that p^m1  n^m1 for each prime p  n
n such that Phi(d,p)  Phi(d,n) for each prime p  n and each divisor d of m.
With m=2, there are plenty examples of the former, but the latter is still unknown. Any n with p1  n1 and p+1  n+1 for each prime p  n must have at least four distinct prime factors (see
paper). If n ≠ 1 mod 24, then n must have at least 5 prime factors. I suppose for higher m, the minimum number of prime factors dividing n will be higher too.