Feuille d'exercices Arithmétique MPSI Lycée Camille Jullian

150
∀n∈Zn4−20n2+ 4
n3p×5qp q
4542
x3+y3+z3= 94 (x, y, z)∈Z
9
9
n p n2+p27n p
7
n un= 34n+2 + 52n+1
un14
2
a b ∀n∈Nun+2 =aun+1 +bun
n∈Zn28 0 1 4
n∈Nn≡7[8] n
n∈Zn2+ 7 |n3+ 5
n∈Nr11n−5
n+ 4 ∈N

n n >6n|(n−2)!
15 15
7 + 8
4+5+6 1+2+3+4+5 1050
x3+x2+ 2x+ 1 = 0 Q
p>2n an=pn−1
b c q r b
c ab∧ac=ac∧ar
ab∧ac=ab∧c
n Fn= 22n+ 1 F
n5 32
Fnn64
F4
Fn+1 = 2 + (Fn−2)Fn
FnFkk1n−1
n6=p Fn∧Fp=Fn∧p
10!
Z
xy = 2x+ 3y
x2+y2−2x+ 4y−5 = 0
x2= 9y2−39y+ 40
1
x+1 y+1 z= 1
9x2−y2= 32
15x2−7y2= 9 3
y3=x2+x
p n vp(n!) =
+∞
X
k=1 n
pk
100!

F0= 0 F1= 1 ∀n∈NFn+2 =Fn+1 +Fn
∀n>1Fn+1Fn−1−F2
n= (−1)n
FnFn+1
∀n∈N∀p∈N∗Fn+p=FnFp−1+Fn+1FpFnFp
FnFn+p
∀(n, m)>2Fn∧Fm=Fn∧m
n>5Fnn
F87Fn
7n8
n S(n)
S(n)n= 32 n= 28 n= 60
S(n)n
S(n)n=pkp k 2
n n =
k
Y
i=1
pαi
ipi
αiS(n) =
k
Y
i=1
pαi+1
i−1
pi−1
m n S(mn) = S(m)S(n)
2 1 3 4
3 4
n3 4
m= 4n−1
5 6
a b
(a+b)∧(a∨b) = a∧b
(a, b)a+b= 144 a∨b= 420
n S(n)
2n
6 28
p2p−1p
n= 2p−1(2p−1)
28
n n = 2a×b b
a>1
S(n) = (2a+1 −1)S(b)S(b)=2a+1c c ∈N
c= 1 b
n2p−1(2p−1) 2p−1

G={a+bi |(a, b)∈Z2}C
(G, +,×)C
G
x y y a b x
y
a b
(p, q)|a−p|61
2|b−q|61 2
z|x−yz|<|y|
z
x y x y y =zx z
x y |x|2|y|2N
7 + 11i1+8i
1G G U
G
U
U
x y (u, v)
ux +vy = 1
x y z x yz x y
x z