A composite number m is called a Carmichael number if the congruence am−1 ≡ 1 (mod m) is true for every number a with gcd(a,m) = 1.
(a) Verify that m = 561 = 3 · 11 · 17 is a Carmichael number. [Hint. It is not necessary to actually compute am−1 (mod m) for all 320 values of a. Instead, use Fermat’s Little Theorem to check that am−1 ≡ 1 (mod p) for each prime p dividing m, and then explain why this implies that am−1 ≡ 1 (mod m).]
(b) Try to find another Carmichael number. Do you think that there are infinitely many of them?
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