Exercice 1 [Nombres de Carmichael]

17
p a p ap−1≡1 mod p
n>4a1< a < n −1
(1) an−16≡ 1 mod n,
n
a2n−2an−1mod n
n
n
n
an−1≡1 mod n a n.
p n p −1n−1
p
n p −1n−1
p2p−1 3p−2p= 3
p≡1 mod 6 p(2p−1)(3p−2)
n
an≡amod n a.
n
n a ∈Z/nZ
n

n=peq p
e>2 (p, q) = 1 a
a≡1 + pe−1mod pea≡1 mod q
p(Z/pZ)∗