Exercice - Alain Camanes

2015/2016
Exercice 1. (-)n∈N,7|52n−4n
Exercice 2. (-)11001011 2 8
16
Exercice 3. ( ! )a, b, c, d ∈Z?a∧b=c∧d= 1
1. a∧d=b∧c= 1 (ac)∧(bd)=1
2. (ac)∧(bd)=(a∧d)(b∧c)
Exercice 4. (-)
1. 37x+ 25y= 1 2. 51x+ 39y= 2004 3. 51x+ 39y= 4
Exercice 5. (-)xx≡3 [11]
x≡4 [15]
Exercice 6. a, b 0< a < b
1. (a+b)∧(a∨b) = a∧b
2. a+b= 144
a∨b= 420.
Exercice 7. (♥)n>0 (n+ 1)|2n
n
Exercice 8. (♥)n>2a∈Zn k ∈Nrk
akn
1. (rk)k∈N
2. 32016 5
3. 13 3126 + 5126
Exercice 9. a, b 0< a < b
1. x, y x2−y2=a2b2
2. x y a = 2 b= 7
Exercice 10. ( ! )n>21
4(n3+ (n+ 2)3)
Exercice 11. (♥)n>2n
Exercice 12. ( ! )N?×N?xy=yx

Exercice 13. (Les nombres de Fermat, ♥)n∈Nn Fn
Fn= 22n+ 1
1 . a) (α, k)∈N22∧k= 1 n= 2αk
b) k>3 2n+ 1 22α+ 1
c) Fnα n = 2α
2. k > 0Fn+k−1=(Fn−1)2kFn+k≡2 [Fn]
3. m6=n FmFn
4. Fn
Exercice 14. (Les nombres de Mersenne, ♥)n∈Nn
MnMn= 2n−1
1. Mnn
2. p2p−1n= 2p−1(2p−1)
3. 2p−1(2p−1) 2p−1