to find all the carmichael numbers N $\displaystyle < 10^{6} $ , we only need to test composite N with primes a from 2 to 19 , and see if they satisfy:
$\displaystyle (a,N)=1 \ \ s.t. \ \ a^{N-1}=1 \ \ mod \ \ N $
to find all the carmichael numbers N $\displaystyle < 10^{6} $ , we only need to test composite N with primes a from 2 to 19 , and see if they satisfy:
$\displaystyle (a,N)=1 \ \ s.t. \ \ a^{N-1}=1 \ \ mod \ \ N $
If prime numbers $\displaystyle a$ satisfy $\displaystyle a^{N-1}\equiv 1(\bmod N)$ then any number $\displaystyle (n,N)=1$ would have to satisfy $\displaystyle n^{N-1}\equiv 1(\bmod N)$. This is because if $\displaystyle a,b$ satisfy this congruence then $\displaystyle (ab)^{N-1}\equiv a^{N-1}b^{N-1}\equiv 1(\bmod N)$. Therefore, there products of all these primes would satisfy the congruence. But since $\displaystyle n$ can be realized as a product of primes it would mean that it itself satisfies the congruence.
You will also find primes as pointed out by ThePerfectHacker, but
how will you distinguish these Carmichael Numbers from primes:
A few Carmichael Numbers < $\displaystyle 10^6$ with smallest factor > 19
252601 = 41 * 61 * 101
294409 = 37 * 73 * 109
399001 = 31 * 61 * 211
410041 = 41 * 73 * 137
488881 = 37 * 73 * 181
512461 = 31 * 61 * 271
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