Feuille de TD n 1 Divisibilité, nombres premiers

◦1
(?)a|b b|c a|c
a b a −b
p
p!+1
1,1,2,3,5,8,13,21,34, . . .
F1= 1, F2= 1 Fn=Fn−1+Fn−2n≥3
1−1
(?)n
2 5 1
1,11,111,1111,...,111 ···111
| {z }
(n+1 )
m m0n
m−m0

n∈Zn3−1
n∈Nn≥2n
√n113 167
(?)a, n ∈N∗\{1}an−1
a= 2 n
a, b ∈ZaZ∩bZ
Zm∈ZmZ=aZ∩bZ
25Z∩10Z
pgcd(56,35),pgcd(309,186),pgcd(1024,729),pgcd(12075,4655)
a b pgcd(a+b, a−b)=1
2
d= pgcd(12075,465,345)
(?)p, q (p > 0, q > 0)
pgcd(p, q)>1
pgcd(p, q)> p
pgcd(p, q)≤min(p, q)
pgcd(p, q)≤ |p−q|
pgcd(p, q)< pq
n∈N∗
pgcd(n, n2+ 1) = 1,pgcd(n2+n, n3+ 1) = n+ 1.
Z2mn =m+n
(a, b)a+b= ppcm(a, b)
pgcd
(?) (a, b)
a+b= pgcd(a, b) + ppcm(a, b)
a, b a +b= 173 pgcd(a, b)

d1= pgcd(56,35)
x y 56x+ 35y=d1
d2= pgcd(309,186), d3= pgcd(1024,729), d4= pgcd(12075,4655).
a b d pgcd
u v
au +bv =d.
u v
(?)d= pgcd(a, b) (u0, v0)∈Z2
au0+bv0=d(u, v)au +bv =d(uk, vk)
k∈Z∗
(uk=u0+kb1,
vk=v0−ka1
a1b1a=da1b=db1
Z2
5x−18y= 4,6x+ 15y= 28.
a, b, c d
(pgcd(a, b) = d)⇐⇒ (pgcd(ac, bc) = dc)
(pgcd(a, b)=1
pgcd(a, c)=1 =⇒(pgcd(a, bc) = 1)
(pgcd(a, b) = 1) ⇐⇒ ∀m≥2,∀n≥2,(pgcd(am, bn) = 1)
(pgcd(a, b) = 1) ⇐⇒ (pgcd(a+b, ab) = 1)
(pgcd(a, b) = d)⇐⇒ ∀n≥2,(pgcd(an, bn) = dn)
(m, n)∈Z24n4+ 1 = 3m2+ 3

n p p|anp|a
(?) 120 = 233151900 = 223252pgcd(120,900)
ppcm(120,900)
720 1400 pgcd(1400,720)
ppcm(1400,720)
(?)n
1n
8 = 2316 = 2432 = 2527 = 33
18 = 213236 = 223272 = 2332
180 = 22325
n=pe1
1pe2 2···per
rpiei∈N
n
a, b ∈N∗ab =cnc, n ∈N∗
a0, b0∈N∗a0n=a b0n=b
a > 0
n
m
(?)y2=x3−x
x x (x−1)/2 (x+1)/2
x x −1x+ 1
n n n + 1
n n n + 2
(y, x)
(?)f:N×N−→ Nf(x, y) =
2x(2y+ 1) −1