Nombres premiers Exercice 1. pgcd × ppcm Exercice 2. pgcd

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×
a, b, c ∈N∗(a, b, c)×(a, b, c) = abc
×
a1, . . . , an∈N∗bi=Q
j6=i
aj
(a1, . . . , an)×(b1, . . . , bn) = (a1, . . . , an)×(b1, . . . , bn) = Qai
ab
a, b ∈N∗ab a b
an=bm
a, b ∈N∗m, n an=bmc∈N∗
a=cmb=cn
52n−1
2 5(2n)−1 2n+2
ar−1
ar−1r a
Mn= 2n−1n
Mn⇒n
M11
an+ 1
a, n ∈Na≥2n≥1an+ 1 n2
n∈N∗dnn
n=ab a ∧b= 1 dndadb
n dn
Q
d|n
d=√ndn
p p ≡ −1 [4]
p n ∧p= 1 np−1≡1 [p]
n∈Np≥3n2+ 1 p≡1 [4]
4k+ 1

1000!
1000 !
1/2+1/3 + ··· + 1/n
n∈Nn≥2xn= 1 + 1
2+1 3+··· +1 n
pn
2qn
pn, qn∈N∗
pn
n
2n2n n
n
2014
1309 1310 1311
n=pqr (1 + 1
p)(1 + 1 q)(1 + 1 r)>2
p≥3n
p n = 2pA[[1, n]]
x∈a2x /∈A
A

a, b, c
a, r 2
a−1|ar−1a= 2 r=pq 2p−1|2r−1r
211 −1 = 23 ×89
M11 = 23 ×89
(−1)(p−1)/2≡1 [p]
Hn⇒H2n⇒H2n+1
2014 = 2 ×19 ×53 3233
1309 = 7 ×11 ×17 1310 = 2 ×5×131 1309 = 3 ×19 ×23
4=22
n=pqr {1, p, q, r, pq, pr, qr, pqr}
(1 + p)(1 + q)(1 + r)
1 + 1
p≤4 31 + 1 q≤6 51 + 1 r≤8 7(1 + 1 p)(1 + 1 q)(1 + 1 r)≤64 35 <2
p= 2 q > 5
q= 3 r
q= 5 r= 7
{2×3×r, 2×5×7}
N1={1,3,4,5,7,9,11,12,13,15,16,}
2
[[1, n]] ∩N1
n= 4 {1,3,4,5,7} {1,5,6,7,8}
A
x∈A\N1xx
2
[[1, n]] ∩N1
A= [[1, n]] ∩N122k×(2l+ 1) 2pxp= 2p−1+
2p−3+··· + 1
p4xp=xp+ 1 + 2p+1 ⇔xp=2p+1 + 1
3p xp=2p+1 −1 3