110: nombres premiers. Applications.

1
N n 6√N Enn
E En
En
a1= 2 Ea1
Eaiai6anan+1
√N
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
√N= 10 3 n= 2
3k+ 2
4k+ 3
6k+ 5
π(x)
x∞
π(x)/x
1/p

pα1
1. . . pαn
npi
pα1
1. . . pαn
np1< p2. . . pn
vp(x)p
x
vp(ab) = vp(a) + vp(b)vp(an) = nvp(a)
Q
p
pvp(x)
(−1)ε(x)pα1
1. . . pαn
n=Q
p
pvp(x)
a b vp(a)6vp(b)
aQ(1 + vp(a))
an|bn
an|bnn61
Z
vp
P1
nn=
1
Σ
Σ
Σ

(p−1)! ≡1[p]
(p!)2≡+−1[p]
p ap≡a[p]
n>2∃a∈Za(n−1) ≡1[n]∀k < n −1, ak6≡ [n]
n
n>2n=pα1
1. . . pαk
k
∀i, ∃ai∈Za(n−1)/pi
i'1[n]∀k < n −1, ak6' 1[n]n
p h ∈ {1, . . . , p −1}n:= 1 + hp2
2n−1'1[n] 2h6' 1[n]n
p= 2127 −1h= 190 1 + 190p2
an−1a= 2
n
2p−1Mp
211−1 = 23 ∗89.
4k+ 3 2p+ 1 Mp
YnY0= 2 Yn+1 = 2Y2
n−1n>3
Mn(2n−1)|Yn−2
23112609 −1
Mp

k
p k Z/pZ−ev n |k|=pn
q=pnq
Xq−XFp
Fq
Fpn⊂Fpmn|m
pFp
Fp
x2−a= 0 Fp
x2−5y2x y
−1K
Z[i]
p= 4m+ 1
(p
q) = pq−1
2
Z/pZ{−1,1}
(2
p)=(−1)p2
−1
8
(p
q)( q p) = (−1)(p−1 2
q−1
2)
F17F1987 Fp
p20k+ 1,3,7,9

p P ∈Z[X]
P=anXn+. . . +a0¯ P=CI(an)Xn+. . . +CI(a0)¯ PZ/pZ
PQ[X]Z[X]
p Xp−X−1Z Z/pZ
X3+ 462X2+ 2433X−67691 Z Z/2Z[X]
X3+ 4X2−5X+ 7 Z Z/2Z[X]
X3−6X2−4X−13 Z Z/3Z[X]
X3+ 30X2+ 6X+ 1 Z Z/5Z[X]
X2+X+ 1 F2
F2(X2+X+ 1)2= 1 + X2+X4
5X4+ 17X3−X2−6X+ 23 Z Z/2Z[X]
X4+ 5X3−3X2−X+ 7 Z Z/2Z[X]
F3
X2+1 X2−X−1X2+X−1
F3
X4+ 7X24X+ 1 Z Z/3Z[X]
Z/2Z[X]
X4+ 4X3+ 3X2+ 7X−4 2,3,5,7et11
P
Z
X4+ 1 FpFp