Efficient Computation Of Full Lucas Sequences

  • Upload
    -

  • View
    218

  • Download
    0

Embed Size (px)

Citation preview

  • 7/22/2019 Efficient Computation Of Full Lucas Sequences

    1/6

    K\MZ^J

    \JKI\@U

    P j e b h c e i f \ j p m r t

    E K { 9 3 3 1 / 5

    Z E F E r y p t m K r m u p P j e b h c e i f \ j p m r t U j r c j s

    b t t p 4 / / w w w . ` c e j . u e f . i e . l j / e r y p t m /

    ^ f i e j ` u F j v i h t , 5

    L - 9 5 0 2 F m u v i c h - f i - H j u v j , L j f k c u o

    ^ b m h j 4 ( + 5 < ) 9 > 0 ; < : 0 9

    A i x 4 ( + 5 < ) 9 > 0 ; 2 1 1 ;

    J e c j h t e m o p u t i t c m h m a a u f f

    F u e i s s j q u j h e j s

    O i r e D m y j i h ` D j i h - D i e q u j s Y u c s q u i t j r

  • 7/22/2019 Efficient Computation Of Full Lucas Sequences

    2/6

    J e c j h t e m o p u t i t c m h m a a u f f

    F u e i s s j q u j h e j s

    O i r e D m y j

    9 )

    i h ` D j i h - D i e q u j s Y u c s q u i t j r

    < )

    D i h u i r y < 3 , 9 3 3 :

    9 )

    @ j p i r t o j h t m a O i t b j o i t c e s ( I K J F ) , Z h c v j r s c t y m a F m u v i c h

    E b j o c h ` u E y e f m t r m h , < , L - 9 5 0 2 F m u v i c h - f i - H j u v j , L j f k c u o

    J - o i c f 4 d m y j N i k j f . u e f . i e . l j

    < )

    @ j p i r t o j h t m a J f j e t r c e i f J h k c h j j r c h k ( @ C E J ) , Z h c v j r s c t y m a F m u v i c h

    ^ f i e j ` u F j v i h t , 5 , L - 9 5 0 2 F m u v i c h - f i - H j u v j , L j f k c u o

    J - o i c f 4 d d q N ` c e j . u e f . i e . l j

    I l s t r i e t . \ j e j h t f y , S j h i h ` F i c b 9 _ p r m p m s j ` i i f k m r c t b o t m q u c e g f y

    e m o p u t j F Z E ` c k c t i f s c k h i t u r j s . P b c s s c k h i t u r j c s l i s j ` m h i s p j e c i f

    t y p j m a t b j F u e i s s j q u j h e j Q

    g

    . C h t b c s p i p j r , w j s b i f f k j h j r i f c z j t b j c r

    o j t b m ` t m i h y t y p j m a F u e i s s j q u j h e j s , i h ` w j s b i f f j x t j h ` c t t m t b j

    ' s c s t j r ' F u e i s s j q u j h e j Z

    g

    . I s i p p f c e i t c m h , w j s b i f f q u c e g f y e m o p u t j

    t b j m r ` j r m a i h j f f c p t c e e u r v j m v j r K A (

    c h t b j j f ` Y (

    p

    @ ) , w b j r j @ ?

    i h a Q

    g

    k

    g >

    w c t b

    p i r i o j t j r s i h Y i r j k c v j h l y

    Z

    g

    ( ^ 7 Y ) ?

    g

    g

    7 ( 9 )

    Q

    g

    ( ^ 7 Y ) ?

    g

    +

    g

    ( < )

    C t e i h j i s c f y l j s b m w h t b i t t b j h u o l j r s Z

    c

    i h Q

    c

    s i t c s a y t b j a m f f m w c h k

    r j f i t c m h s

    Z

    c + d

    ? Z

    c

    Q

    d

    Y

    d

    Z

    c d

    7 ( 5 )

    E K { 9 3 3 1 / 5

  • 7/22/2019 Efficient Computation Of Full Lucas Sequences

    3/6

    J e c j h t e m o p u t i t c m h m a a u f f F u e i s s j q u j h e j s

    , w b j r j

    G

    d

    ?

    h 9

    R

    c ? d

    g

    c

    7 9 k i h g

    h 9

    ? 9

    B j h e j ,

    G

    d 9

    ? g

    d 9

    +

    ? 9 ) , Z

    g

    ( ? Z

    f

    >

    ) ? Z

    b

    9

    Q

    f

    9

    Y

    f

    9

    . P b u s , m h f y

    t b j v i f u j m a Z

    b

    9

    c s h j j ` j ` . [ j m h f y h j j ` t m s b m w t b i t t b j v i f u j m a Z

    b

    d 9

    e i h

    l j ` j r c v j ` a r m o Z

    b

    d

    . L y ( : ) i h ` ` j p j h ` c h k m h t b j v i f u j m a g

    d 9

    , w j b i v j t b j

    a m f f m w c h k e i s j s 4

    c a g

    d 9

    ? > , t b j h ( f

    d 9

    7 b

    d 9

    ) ? ( < f

    d

    7 f

    d

    + b

    d

    ) 7

    c a g

    d 9

    ? 9 , t b j h ( f

    d 9

    7 b

    d 9

    ) ? ( f

    d

    + b

    d

    7 , t b j h b

    d 9

    ( ? b

    d

    + f

    d

    ?

  • 7/22/2019 Efficient Computation Of Full Lucas Sequences

    4/6

    J e c j h t e m o p u t i t c m h m a a u f f F u e i s s j q u j h e j s 5

    C h p u t s 4 g ? , t b j h t b j

    e m o p u t i t c m h m a Z

    g

    i h Q

    g

    r j q u c r j s

    9 9 h

    , 5 : , ( 9 : 9 ) , p p . 9 > > 5 {

    9 > < 1

    ; o j h j z j s , i . d . 4 n J f f c p t c e e u r v j p u l f c e g j y e r y p t m s y s t j o s ' ( G f u w j r I e i -

    ` j o c e ^ u l f c s b j r s , 9 3 3 5 )

    2 o j h j z j s , i . , q u , o . , i h v i h s t m h j , s . 4 n @ r i a t m a C J J J ^ 9 5 1 5 , e b i p -

    t j r 1 ' , H m v j o l j r 9 3 3 :

    E K { 9 3 3 1 / 5