Sujet 13

Z/nZ
x y n xy
x+y> n
xy
x+y≥n+1 n2+2n+2
n5n−1+ 3n−15n+ 3n
n an=n3+ 1 dn=P GCD(an, an+1)
dn
n+ 1 2n
n > 1 2n+ 1 n
x, y, z x3+ 9y3= 3z3
x, y y2=x3+ 16
x, y x2+ 2y2= 3z2

x, y x2=y5−4
79999
a, b, c 2c−1 2a+ 2b+ 1
a, aa, aaa, ... n
xx7−1
x−1
x, y x7−1
x−1=y5−1
p q ≥p
(q!)p−1
q!−1p
p
p
−3p
p E =Z/pZ\ {0,1}f:E→E, x 7→ 1−x−1
f f
E
f p −2
x2−x+ 1 = 0 Z/pZ
p
a p ap−1
3
p
b b2+b+ 1 ≡0 mod p
−3p
R[X, Y ]Z[X]Z/23Z[X]Z/4Z[X]
G n > 1|G| ≤ nlog2n
m∈N∗n∈N−{0,1}k
GLk(C) (Z/nZ)m
Z[j]
(Z/2nZ)×
p a ∈Za
p a(p+1)/2≡amod p

K|·|:K→R+
x∈K|x|= 0 x= 0 x, y ∈K|xy|=|x||y| |x+y| ≤ |x|+|y|
|·| x, y ∈K|x+y| ≤ max(|x|,|y|)
|·| n∈N|n·1| ≤ 1
Q
Sn→C×
Snn!
2
nSn
An+2 GLn(k)k
p, q q|2p−1q≡1
mod 2p
k(k, +) (k×,×)
α∈CkC
k[α]α k
k
Q[√2] Q[3
√3] Q
K=Q[√2][ 3
√3]
KQ
(1,√2,3
√3,3 √9,6 √72,6 √648) Q
KQ