Exercice 1. - Université de Rouen

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[•]
a b
a= 254 b= 12
a=−254 b= 12
a= 254 b=−12
a=−254 b=−12
[•]
a6= 1 n a −1an−1
n n + 1 n2+ 1
[•]
a b p ab
p a b
a b c
a bc a b a c
a b c ab c
a b c a bc
a b a +b ab
[•]
a n a ≥2n≥2
an−1a= 2 n
n∈N
6 5n3+n[•]
9 4n−1−3n
16 5n−1−4n
N2x+y= 187 (x, y) = 30 (x, y)
[•]
n∈N
5 22n+1 + 32n+1 [•]
7 32n+1 + 2n+2
11 3n+3 −44n+2

n(n3−n)(58n+4 + 34n+2)
3804
(3123 −5) 25 [•]
(2443 + 7) 15
N2(x, y) = 12 (x, y) = 216
a p
k1≤k≤p−1pp
k
a p ap−a
n n2≡1 [8]
n n2≡0 [8] n2≡4 [8]
a2+b2+c2a b c
(x, y)∈Z2
xy = 2x+ 3y
x2−y2−x+ 3y= 30
2x3+xy −7=0
31000 7
n3n7
n3n+6 −3n7
n sn=1+3+.... + 3n−1
sn=1
2(3n−1)
n sn7
7n+ 1 8 n
8 7n+ 1 n

n∈N40n·n! (5n)!
1665 1035
(un)n∈Nu0= 14 un+1 = 5un−6
u1, u2, u3, u4, u5
u0≡u2[4] u2≡u4[4] u3≡u1[4] u5≡u3[4]
un+2 un
∀n∈N, un+2 ≡un[4]
∀k∈N, u2k≡2 [4] u2k+1 ≡0 [4]
∀n∈N,2un= 5n+2 + 3
∀n∈N,5n+2 ≡25 [100]
∀n∈N,2un≡28 [100]
∀n∈N,(un≡14 [100] un≡64 [100])
∀k∈N,(u2k≡14 [100] u2k+1 ≡64 [100])