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