Semaine 12

Z,+
p
pp
kk∈ {1,2, . . . , p −1}
a b (a+b)p−ap−bpp(a+ 1)p−(ap+ 1) p
a a ap−a p
Pn(x) = anxn+an−1xn−1+· · · +a0aiPnr/s r s
s anr a0
(Fn)n∈NF0= 0 F1= 1 n∈N
Fn+2 =Fn+1 +Fn.
n≥1FnFn+1
m < n Fn=Fm+1Fn−m+FmFn−m−1m n
m≤n FnFmFnFn−m
FmFnFdd m n
a, b, c b a c a b c bc a
σN N
∀n∈Nσ(n) =
d|n
d.
n σ(n) = 2n
6 28
σ(p)σ(pk)p k ∈N
σ(pkql) = σ(pk)σ(ql)p q k, l ∈Nσ(mn) = σ(m)σ(n)n m
n= 2p−1(2p−1) p2p−1n
n n = 2tm t m
2t+1m= (2t+1 −1)σ(m)m= (2t+1 −1)M M ∈N
σ(m) = m+M M = 1 n
Z,+
p
pp
kk∈ {1,2, . . . , p −1}
a b (a+b)p−ap−bpp(a+ 1)p−(ap+ 1) p
a a ap−a p
Pn(x) = anxn+an−1xn−1+· · · +a0aiPnr/s r s
s anr a0
(Fn)n∈NF0= 0 F1= 1 n∈N
Fn+2 =Fn+1 +Fn.
n≥1FnFn+1
m < n Fn=Fm+1Fn−m+FmFn−m−1m n
m≤n FnFmFnFn−m
FmFnFdd m n
a, b, c b a c a b c bc a
σN N
∀n∈Nσ(n) =
d|n
d.
n σ(n) = 2n
6 28
σ(p)σ(pk)p k ∈N
σ(pkql) = σ(pk)σ(ql)p q k, l ∈Nσ(mn) = σ(m)σ(n)n m
n= 2p−1(2p−1) p2p−1n
n n = 2tm t m
2t+1m= (2t+1 −1)σ(m)m= (2t+1 −1)M M ∈N
σ(m) = m+M M = 1 n