Mersenne 2

1
n p 16p6n q
n=pq
p > √n q < √n6p q p n
p
p6√n
37901 37901 = 151 ×251
36 151
37907 44
a≡1 (mod (a−1)) an≡1 (mod (a−1))
an+1 −1 = a×(an−1) + a−1
1 + a+a2+··· +an−1=an−1
a−1
2013 −1 20 −1 = 19
235 −1 235 −1 = 257−1 25−1 = 31
27−1 = 127 235 −1 = 275−1
an−1a−1
a > 2n>2 1 < a −1< an−1an−1
n=pq 2n−1 = (2p)q−1 2p−1
n p n =pq 1< p < n
1<2p−1<2n−1 2n−1
4 6 8 9 10 12 14 15 16
M2= 3 M3= 7 M5= 31
M7= 127
M11 = 2047 = 23 ×89 23
M13 = 8191 24

341 2340 (mod 341)
22≡4 (mod 341)
24≡16 (mod 341)
28≡256 (mod 341)
216 ≡64 (mod 341)
232 ≡4 (mod 341)
264 ≡16 (mod 341)
2128 ≡256 (mod 341)
2256 ≡64 (mod 341)
340 = 256 + 64 + 16 + 4 2340 ≡64 ×16 ×64 ×16 ≡1 (mod 341)
2340 ≡1 (mod 340) 341
k0ak0≡1 (mod p)
k k0k=q×k0+r06r < k0
ak≡aq×k0+r≡ak0q×ar≡ar
ak≡1 (mod p)ar≡1 (mod p) 0 6r < k0k0
r= 0 k k0
p q q Mp2p≡1 (mod q)
2q p p 2p
q q 6= 2 2q−1≡1 (mod q)
q−1p2q
q−1k q −1=2kp q = 2kp + 1 
M11 23 89 2k×11 + 1
M17 = 131 071 4
2k×17 + 1
M19 = 524 287 7
2k×17 + 1
M23 47 = 2 ×23 + 1
M29 233 = 8 ×29 + 1
M31
q M31 q248k+ 1 248k+ 63
M31

a≡ −1 (mod (a+ 1)) an≡ −1 (mod (a+ 1)) n
1−a+a2+··· − an−1=1−(−a)n
1 + a=an+ 1 a+ 1 n
63+ 1 6 + 1 = 7
212 + 1 212 + 1 = 243+ 1 24+ 1 = 17
m=pq p > 1q < m
am+ 1 = (aq)p+ 1 aq+ 1 1 < aq+ 1 < am+ 1 am+ 1
F0= 3 F1= 5 F2= 17 F3= 257
p
p Fn22n≡ −1 (mod p) 22n+1 ≡1 (mod p)
2p2n+1
2n+1 2k06k6n+ 1
2k06k6n22n≡1 (mod p)
2p2n+1
2p−1≡1 (mod p)
p−1 2n+1 k p =k×2n+1 + 1 
F4= 65 537 2
k×25+ 1
F5k×26+ 1 = 64k+ 1
F5= 4 294 967 297 = 641 ×6 700 417
Fn
7
64k+ 1

q Mpp q ≡ ±1 (mod 8)
M37 M41 M43 M47 M53
3050 ×59 + 1 = 179951 M59
M59/179951
M61
Fn+1 −2 = Fn×(Fn−2)
Fn+1 −2 = Fn×Fn−1× ··· × F1×F0
p Fnn>2k p =k×2n+2 + 1
Fn−12n+2 p
Fnn>2k×2n+2 + 1 k
F6
F6= 274 177 ×67 280 421 310 721