MA14 U16

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Transcript of MA14 U16

  • 7/22/2019 MA14 U16

    1/8

    n! n

    ba

    b a

    a > b

    b

    a

    = 0

    a N b, c N

    a

    b

    c

    = a

    bc

    .

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    q1 =a

    b

    e q2=

    ab

    c

    .

    a= bq1+r1, com r1 b 1

    ab

    = q1 = cq2+r2, com r2 c 1.

    a= bq1+r1= b(cq2+r2) +r1= bcq2+br2+r1.

    br2+r1 b(c 1) +b 1 =bc 1,

    q2 a bc

    q2= a

    bc

    .

    c a b

    a b c

    p m

    Ep(m) p m

    p m

    Ep(n!)

    p

    n!

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    a m a >1

    n

    an

    > m

    A= {an; n N}

    an m n N A

    r N x ar x A a 2

    ar+1 2ar > ar,

    ar+1 A ar A

    n p

    Ep(n!) =

    n

    p

    +

    n

    p2

    +

    n

    p3

    +

    r pi > n i r n

    pi

    = 0 i r

    n

    n= 0

    m m < n p n

    p, 2p, . . . ,

    n

    p

    p.

    Ep(n!) =

    n

    p

    +Ep

    n

    p

    !

    .

    Ep

    n

    p

    !

    =

    n

    p

    p

    +

    n

    p

    p2

    +

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    Ep(n!)

    n= pq1+r1

    q1 = pq2+r2

    . . .

    qs1= pqs+rs

    q1 > q2 > s qs < p

    Ep(n!) =q1+q2+ +qs.

    10!

    Ep(10!)

    p 10 E2(10!) = 5 + 2 + 1 = 8, E3(10!) = 3 + 1 = 4, E5(10!) = 2

    E7(10!) = 1

    10! = 2834527.

    10! 10!

    a1, . . . , am, b b= 0 a1+ +am

    b

    a1b

    + +

    amb

    .

    qi ri

    ai b i = 1, . . . , m

    ai= bqi+ri

    a1+ +am= (q1+ +qm)b+r1+ +rm.

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    a1+ + am b

    q1+ +qm

    r1+ +rm

    b 1

    a1, . . . , am, b b = 0

    (a1+ +am)!

    a1! am! .

    p

    i

    a1+ +am

    pi

    a1

    pi + +

    am

    pi .

    Ep((a1+ +an)!) Ep(a1!) + +Ep(am!),

    Ep(n!) p n

    p

    p, n N p

    n= nrpr +nr1pr

    1 + +n1p+n0

    p n

    Ep(n!) =n (n0+n1+ +nr)

    p 1 .

    0 ni< p n

    p

    = nrp

    r1 + nr1pr2 + + n2p + n1

    n

    p2

    = nrpr2 + nr1pr3 + + n2

    . . .n

    pr

    = nr

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    Ep(n!) =n

    p

    + n

    p2

    + + n

    pr

    =

    nrpr 1

    p 1 +nr1

    pr2 1

    p 1 + +n1 =

    nrpr +nr1p

    r1 + +n1p+n0 (nr+nr1+ +n1+n0)

    p 1 =

    n (n0+n1+ +nr)

    p 1 .

    100!

    2 5 10000!

    10000!

    10000!

    n 5

    n! 584

    n 37

    n!

    a1, . . . , am N b N

    a1b

    + +

    amb

    a1+ +am

    b

    a1b

    + +

    amb

    +m.

    m, n N (m, n) = 1

    (m+n 1)!m!n!

    N.

    m,n,b N b= 0

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    2m

    b

    +

    2n

    b

    m

    b + n

    b +

    m+n

    b

    (2m)!(2n)!

    m!n!(m+n)!

    n, m N (n m)! [(n!)m, (m!)n]

    (n!)(n1)! (n!)!

    n, a1, . . . ar N d = (a1, . . . , ar)

    d(n 1)!

    a1! ar!.

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    1 +1

    4+

    1

    9+

    1

    16+

    1

    25+

    2/6