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Old 2020-06-30, 19:28   #10
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Nov 2016

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Originally Posted by Dr Sardonicus View Post

The ne plus ultra of "generalized Carmichael numbers" appears to be described in a paper entitled Higher-order Carmichael numbers by Everett W. Howe.
There appear to be two types of Higher order Carmichael numbers of order m:

n such that p^m-1 | n^m-1 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 p-1 | n-1 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.

Last fiddled with by carpetpool on 2020-06-30 at 19:36
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