MA14 U14
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Transcript of MA14 U14
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n
2m
Fn= 22n + 1.
F1 = 5, F2 =
17, F3= 257, F4 = 65537
F5 = 225 + 1 = 4.294.967.297 = 641 6700417,
(Fn, Fm) = 1, n=m.
a n an 1
a= 2 n
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an 1 a >1 n >1
a > 2
a1 > 1
a1|an
1 an 1
a= 2
n
n= rs r >1 s >1 2r1 (2r)s1 = 2n1
2n 1
n
Mp = 2p 1,
p
2 p 5000
p
M13466917
3, 7, 11, 15, . . . , 3 + 4n , . . .
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4n+ 3
4n+ 1 4n+ 3
={4n+ 1; n N}
(4n+ 1)(4n + 1) = 4(4nn +n+n) + 1.
p1 < < pk 4n+ 3 n 1
a = 4(p1 p2 pk) + 3
3, p1, . . . , pk
4n+ 1 a 4n+ 1
4n+ 3
4n+ 1
x N x 2 x2 + 1
4n+ 1
p = 2
x2 + 1 4n+ 1
={4n + 1; n N}
p|x2 + 1 p
(p1)/2 N N x2 + 1 =p
x2 =p 1.
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(p1)/2
,
N
xp1 = (x2)p1
2 = (p 1)p1
2 =
p+ 1, p12
p 1, p12
xp1 =p 1,
xp1 1 =p 2.
p|x2+1 p|x
p|xp1 1 p|2
p 1
2
p 4n+ 1
1, 5, 9, 13, 17, . . . , 4n+1, . . .
p1, . . . , pk
4n+ 1
a= 4p21 p2
k+ 1.
pi | a i = 1, . . . , k a
4n+ 3
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n, a N
m N (a+ 1)n =ma+ 1
a >0 m N (a 1)2n+1 =ma 1
a >1 m N (a 1)2n =ma+ 1
n
Fn 4m + 1
p
q
(Mp, Mq) = 1.
n, m N
m < n Fm|Fn 2
(Fn, Fm) = 1 n=m
6n+ 5
3n+ 2
pn n pn 22n2 + 1
(Fi, Fj) = 1 i = j 2 | Fi F5
(un) n
un 4k+ 1
u2n un1un+1+ (1)n+1
n
u2
n un1un=u2
n1+ 1
un u2n1+ 1
