Exercices d`arithmétique supplémentaires

a, b s =a+b p =ab
p
s= 2 s= 3
p p 6= 1 p6= 2 p= 136 s≥22
p
s6= 2 s6= 3
s p =ab
s−1
s
d+p/d −1d p
s−1 = 135 d= 1 1 + p−1p=s−1 = 135
d+ 135/d −1d135
135
d+ 135/d −1
[x]x
1
2011+2 2011+22 2011+... +22009 2011.
p
2p p ≡ ±1 mod 8
2011 ≡3 mod 8 2 2011
21005 ≡ −1 mod 2011 2011|2i(21005 +1) = 2i+1005 +2i
2i21005+ib2i
2011 c+b2i+1005 2011 c=
2i+2i+1005
2011 −1i
22010−1
2011 −1005
p a p ap−1
2≡1 mod p

a p ap−1
2≡ −1 mod p a
p2p
p≡ ±1 mod 8
29 = 22+ 522×29 = 32+ 72n
n n
n= 2r1+...2rk+ 2s1+... + 2slri
si
r1< r2< ... < rk
s1< s2< ... < slr= 2r1+... + 2rks= 2s1+... + 2sl
4×29r= (32+ 72)×2r=Pk
i=1(3 ×2ri+1
2)2+ (7 ×2ri+1 2)2
4×29s= (22+ 52)×4s=Pk
i=1(4 ×2si
2)2+ (10 ×2si 2)2
116n= 4 ×29n= 4 ×29(r+s)
n n n =
116q+t0≤t < 116 r≡Pr−1
k=0(58k+ 1)2( mod 116)
n > P114
k=0(58k+ 1)2n−Pr−1 k=0(58k+ 1)2
Pr−1
k=0(58k+ 1)2
n