Nombres Carmichael - Agreg

p
n= 4 34= (−1)4= 1[4]
561
n
n
n∀p∈ P|n, p −1|n−1
n
n=pqr p < q < r
q r p
n
p p2|n pn=p[n]
p2|pn−p p2|p n ≥2n n =
p1· · · pspiZ/piZ∗
bi∈Z/piZ∗0bipi−1Z/piZ∗
b∈Z/nZb=bi[pi]i bn=b[n]n bn
i=bi[pi]
i pi−1|n−1i
n p n p −1|n−1
n=p1· · · pspia i api−1= 1[pi]pi
a an=a[pi]pi−1|n−1an=a[pi]an=a[n]n
n(−1)n= 1[n]n≥3
n=pq p < q q −1|n−1n−1 = pq −1 =
(p−1)q+q−1q−1|p−1q−1q q −1≤p−1
n
q−1n−1 = pqr −1 = (pr −1)q+ (q−1) q−1|pr −1
r−1|pq −1 (q−1)(r−1)
(pr −1)(pq −1) −p2(r−1)(q−1) = 1 −pr −pq −p2(1 −r−q) = p2(r+q)−pr −pq + 1 −p2
(q−1)(r−1) ≤p2(r+q)≤2p2r qr ≤2p2r+r+q−1≤2r(p2+ 1) q≤r

r q ≤2(p2+ 1) r−1≤pq −1r≤pq ≤2p(p2+ 1) r q
p
Z/nZ
pZ/pZ∗
O(d)Z/pZ∗dPd|p−1|O(d)|=p−1
d|p−1O(d)x∈Z/pZ∗x d hxi
Z/pZXd−1Xd−1
dhxi
y∈O(d)y∈ hxiy Xd−1hxi ∼ Z/dZ
ϕ(d)|O(d)| ≤ ϕ(d)
X
d|p−1
|O(d)| ≤ X
d|p−1
ϕ(d) = p−1
|O(d)|=ϕ(d)d p −1O(d−1)
Z/pZ∗