Théorème des deux carrés

p
p
p≡1[4]
p1 4
(Z/pZ)∗
n∈Zn2≡ −1[p]
(a, b)∈Z2
0<√b < √p¯
¯
¯
¯
bn
p−a¯
¯
¯
¯≤1
√p
p= (bn −ap)2+b2
4 0 1 4 0 1
2
p0 2 2
1 4
pK=Z/pZx, y ∈K∗
x2=y2⇔(x−y)(x+y) = 0 ⇔x=y x =−y
K∗K∗x7→ x2
x∈K∗x6=−xKp > 2Card(K∗)
2=p−1 2
K∗
−1Kx∈K∗x=y2y∈K∗
x(p−1)/2=y2(p−1)/2=yp−1= 1
x P =X(p−1)/2−1p−1
2KP P
p−1
2KP
K∗
p≡1[4] p−1
2(−1)(p−1)/2= 1 −1K
N=E(√p)+1 ξ=n
pN xk=kξ −E(kξ) [0,1[ 0 ≤k≤N−1
N·0,1
N·,·1 N,2 N·, . . . , ·N−1 N,1·
•xk·N−1
N,1·x0= 0 k > 0b=k
a=E(kξ)+1
0< b ≤N−1<√p¯
¯
¯
¯
bn
p−a¯
¯
¯
¯
=|xk−1| ≤ 1
N≤1
√p
N−1<√p√p6∈ N

•N xkN−1·k
N,k+ 1 N·0≤k≤N−2
k l xkxl
k < l b =l−k a =E(lξ)−E(kξ)
0< b ≤N−1<√p¯
¯
¯
¯
bn
p−a¯
¯
¯
¯
=|(l−k)ξ−(E(lξ)−E(kξ))|=|xl−xk| ≤ 1
N≤1
√p
(a, b)∈Z2
0< b < √p¯
¯
¯
¯
bn
p−a¯
¯
¯
¯≤1
√p
0< b2< p (bn −ap)2≤p0<(bn −ap)2+b2<2p
n2+ 1 ≡0[p]
(bn −ap)2+b2=b2(n2+ 1) −2admp +a2p2≡0[p]
]0,2p[
(bn −ap)2+b2=p
1
4