Feuille TD 2-HLMA304-arithmétique-2016-2017 Fichier

◦2
P∗2
RP∗
∀(p, q)∈(P∗)2,p+q
2∈ P∗
(?)E x E
∼ P(E)
E
∀A, B ∈ P(E), A ∼B⇐⇒ A=B
∀A, B ∈ P(E), A ∼B⇐⇒ A⊂B
∀A, B ∈ P(E), A ∼B⇐⇒ x∈A∪B
∀A, B ∈ P(E), A ∼B⇐⇒ (x∈A∪Bou x∈A∩B)
(?)∼Zx∼y⇐⇒ x2≡y2[5]

a b 5
a2+b2a2−b25
(?)n, n + 2, n + 4
3
p p + 20 p+ 22
−1 8
(?) 9 9
a3+b3+c33abc
(m, n)∈Z2n6−n3= 7m2+3
7
p m, n ∈Zm2≡n2[p]
m≡n[p]m≡ −n[p]
m n m2≡n2[8] m6≡
±n[8]
m2≡n2[6] m≡ ±n[6]
210 = 1024 210n
25
2123456 24≡1[5]
(?) 3123456789 32≡ −1[10]
123456789 ≡1[4]
? n ∈N∗G=Z/nZk∈Zd= pgcd(k, n)
k a ∈N∗a.k =k+· · · +k
| {z }
a
= 0
k G n/d
k d G

? a, b G
a m
b n
pgcd(m, n)=1
ab =ba.
ab
? G H
G G
∀x, y ∈G, x ∼y⇐⇒ ∃ h∈H x =hy.
∼e
a∈G H G h ha
H a
Card HCard G
Z/nZ
n
Z/nZ0
?Z/1024Z
Z/5Z,Z/10Z,Z/11Z,Z/24Z,Z/25Z,Z/36Z.
Z/10Z
x2= 1 Z/5Z(x−1)(x+
1) = 0
x2= 1 Z/10Z
(x−1)(x+ 1) = 0 Z/10Z
(x−1) ∈ {0,5}(x+ 1) ∈ {0,5}
x
x2= 1 Z/10Z

? x(x+ 1) Z/10Z
x2=−1
Z/11Z
? p x2= 1
Z/2pZ1−1
p, q ∈N\ {0,1,2}pgcd(p, q)=1 x2= 1
Z/pqZ1−1
??
55555
?? y2=x5−4
x≡1[3]
x≡3[5]
x≡4[7]
x≡2[11]
,
x≡997[2001]
x≡998[2002]
x≡999[2003]
10100 13 19
10100 247 = 13 ×19 1099 + 1
247
(x≡9[14]
x≡13[31]
? n ∈N∗ϕ(n)
(Z/nZ,×)
ϕ(p)ϕ(pα)p α ∈N∗
m n f :Z/mnZ−→
Z/nZ×Z/mZf(x) = (ˆx, ˜x)f
ϕ(mn) = ϕ(m)ϕ(n).
ϕ(n)n