Planche C

C
p p −1 = 2st t a
p
at≡1 (mod p)
∃i, 0≤i < s, a2it≡ −1 (mod p)
n n −1=2st t
a1< a < n at6≡ 1 (mod n)i∈[0, s −1] a2it6≡ −1
(mod n)n a
n > 9n−1=2st t
a
φ(n)
561
n a
Z/nZn−1n
(a, n) = 1 d a
n d|φ(n)q1, q2, . . . , qrr
n−1
[∀i∈[1, r], ap−1
qi6≡ 1 (mod n)] ⇒[n].
d n−1xd≡1 (mod n)
i∈[1, r]d=p−1
qi
(u, v, 1),∀(u, v)∈N×N
1 (n, a, r)`(n, a, rq)q a n−1
q≡1 (mod n)
2 (n, a, n −1) `n ap−1≡1 (mod n)
n n 2
3n n −1
d4 log2ne+ 1
nd4 log2ne − 4
2 3
xn2blog2nc
Z/mZ