Feuille d`exercices n 1

◦
a a > 2n > 1an−1
2n−1n
n∈NFn= 22n+ 1
2n+ 1 n2
5×27+ 1 54×228 −1 54+ 2454×228 + 232 F5
(m, n)FnFm
n>m 22n+ 1 = (22m)2n−m−(−1)2n−m+ 2
x2+y2=z2N∗
(x, y, z)x y z
x y x
z−y
2
z+y
2
R2O= (0,0)
A B R2det(−→ OA, −−→ OB)
O, A, B, O +−→ OA +−−→ OB A, B ∈Z2
A B R2]A, B[
Z2]A, B[∩Z2=∅)
A= (a, a0)∈Z2\ {O}A O a a0
A= (a, a0)B= (b, b0)Z2O O
P∈ M2(Z)
a
a0=P1 0  c c0=P−1b b0C= (c, c0)∈Z2I= (1,0) ∈Z2
Z2O, A, B, O +
−→
OA +−−→ OB Z2
O, I, C, O +−→ OI +−−→ OC
|det(−→ OA, −−→ OB)|Z2
O, A, B, O +−→ OA +−−→ OB
A= (a, a0)∈Z2O M = (m, m0)∈Z2
am0−ma0=±1.
n(n−1)! n
n
n= 4
Z
x≡2 (mod 7)
x≡3 (mod 11) x≡4 (mod 21) x≡10 (mod 33)
77777777
9002000 101102103
313233
100100100

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Z2212x+ 171y= 1
Z/212Z171x= 7
p a ∈Z/pZx∈Z/pZa=x2
k p k Z/pZ
p p ∈ {3,5,7,11,13}.
1
2(p−1) (Z/pZ)∗
ϕ: (Z/pZ)∗→(Z/pZ)∗ϕ(x) = x2
ε: (Z/pZ)∗→+1,−1ε(a) = a(p−1)/2
k k p k(p−1)/2≡1 (mod p)
p−1p
(k
p) 1 k p −1
x2+ 4x−1 = 0 Z/11Z
x2+ 5x+ 2 = 0 Z/11Z
x2+ 6x−13 = 0 Z/21Z
x2+ 3x+ 2 = 0 Z/6Z
x2+ 4x+ 6 = 0 Z/9Z
p q n =pq c ϕ(n) (n, c)
d cd ≡1 (mod ϕ(n)) d
a∈Zacd ≡a(mod n)