Ευκλειδης Β 60

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Ευκλειδης Β 60

Transcript of Ευκλειδης Β 60

  • rg() f g'(x)f(g(x))dx= J_ f(x)dx g()

    Jg(x) , . f(t)dt = f(g(x))g'(x)

    J

    J f '(x)g(x)dx= [t(x)g(x) J - f(x)g'(x)dx

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  • ETAIPEIA 60 2006 : 3,50

    e-mail: [email protected] www.hms.gr

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    ..................................................... 17 ................................................ / ...... . . . . ..... 25

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    ' ./ y ./ ./

    ..... .. . 31

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    t.: , ... " "''. e t:. 3,50 Q (10;00 + 4,00 Q: Q 14,00) 10,00

    ' .w. . ... . 54 .. 30044 t ...

  • :

    , , , . . t , , Mersenne. AJJ..f:. ' . m ' , , n

    , . , /, mo

    . , , ,

    , .. . . , . Fermat, :

    = 22" + 1 , n=0,1,2,3,4. Fermat Toicelli .

    Piee de Fenat 1601- 1 665

    Euler, , . . .. Mersenne (. ), (. 2 , . ' ).

    .

    25 2005 , :

    (i) (http://www.mersenne.org/30402457.htm) (ii)(http://mathword. wolfram.co/news/2005-12-25/

    mersenne-43) Great Intemet Mersenne Pme Search (GIMPS,

    www.mersenne.org) Curtis Cooper Steen Boone Central Missou State Uniersity, ( Lucas- Lehmer) 15 2005 230.42.457 1 = 31541647561884608093 . . . 11134297411652943871

    9.152.052 , 43 Mersenne .

    Mersenne :

    ,

    (1) p

    2, (. )

    Mersenne Lucas 1644, Marin Mersenne

    (1588 - 1648)

    ' ' .4/2

  • ------------- -------------

    , Cogitata physica mathematica, p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 257 44 p 257, . : 67 257 61, Ms9 .

    Mersee (Ree Descartes, 1596-1650) Fermat (Piee de Fermat, 1601-1665), (Galileo Galilei, 1564-1642) Huygens (Christiaa Huyges, 1629-1695).

    , Mersee , Lucas- Lehmer. , . Franois Edouard Anatole Lucas ( 1842 - 1891), ( ) { Fibonacci (Leoardo Pisao Fiboacci, 1170 - 1250), Lucas, ). Lucas, , 49 . , Derrick Henry Lehmer (1905 - 1991)

    D. H. Lehmer D.N. Lehmer

    Berkeley. Derrick Norman Lehmer (1867 - 1938). , , -

    . Emma Markovna Trotskaia Leh

    mer (1906 - ) ( ) ( ... ) ( ). , ( ). a Berkeley. .

    Lucas - Lehmer ( LL) :

    (Sn) : Sn + = (S0)2- 2, Sz = 4, = 2, 3, ...

    Mersene = 2 - 1 p > 2 S . Sn mod = 2, 3, ... , p.

    GIMPS Mersee. .

    Mersee (. c W. Weisstei, http://mathword. wolfram.com/ews/2005-12-25/mersee-43). ? , .. ( LL) 6972593 38 Mersee 13466917 Mersee ( 39?). ??

    - -/

    1 2 1 2 3 1 3 5 2 4 7 3

    ' ' .4/3

  • ------------- -------------

    5 13 4

    6 17 6

    7 19 6

    8 31 10

    9 61 19

    10 89 27

    11 107 33

    12 127 39

    13 521 157

    14 607 183

    15 1279 386

    16 2203 664

    17 2281 687

    18 3217 969

    19 4253 1281

    20 4423 1332

    21 9689 2917

    22 9941 2993

    23 11213 3376

    24 19937 6002

    25 21701 6533

    26 23209 6987

    27 44497 13395

    28 86243 25962

    29 110503 33365

    30 132049 39751

    31 216091 65050

    32 756839 227832

    33 859433 258716

    34 1257787 378632

    35 1398269 420921

    36 2976221 895832

    37 3021377 909526

    38? 6972593 2098960

    39?? 13466917 4053946

    40?? 20996011 6320430

    41?? 24036583 7235733

    42?? 25964951 7816230

    43?? 30432457 9152052

    . (325; ..- 265; ..) :

    = 2-1 (2) , .

    , Leonhard Euler (1707 - 1783) . (2). , -

    1461 Regius, Cataldi

    1588 Cataldi

    1588 Cataldi

    1750 Euler

    1883 epaywHH, Seelhof

    1911 Powers

    1913 Powers

    1876 Lucas

    30/1/1952 Robinson

    30/1/1952 Robinson

    25/611952 Robinson

    7/10/1952 Robinson

    9/10/1952 Robinson

    8/9/1957 Riesel

    3/11/1961 Hurwitz

    3/11/1961 Hurwitz

    11/5/1963 Gillies

    16/5/1963 Gillies

    2/6/1963 Gillies

    4/3/1971 Tuckerman

    30/10/1978 Noll & Nickel 9/2/1979 Noll

    8/4/1979 Nelson & Slowinski 25/9/1982 Slowinski

    28/1/1988 Colquitt & Welsh 20/9/1983 Slowinski

    6/9/1985 Slowinski

    19/2/1992 Slowinski & Gage 10/1/1994 Slowinski & Gage 3/9/1996 Slowinski & Gage

    12/11/1996 Joel Armengaud/GIMPS

    24/8/1997 Gordon Spence/GIMPS

    27/1/1998 Roland Clarkson/GIMPS

    1/6/1999 Nayan Hajratwala/GIMPS

    14/11/2001 Michael Cameron/GIMPS

    17/11/2003 Michael Shafer/GIMPS

    15/5/2004 Josh Findley/GIMPS

    18/2/2005 Mai'tin Nowak/GIMPS

    15/12/2005 C. Cooper & S. Boone/GIMPS

    43 : 2 = 6, 3 = 28, ... ' 30402457 -y ;

    ( Mersenne). , . , , y: 300 , 75 -

    ' ' .4/4

  • ------------- -------------

    10000019, 10007 101 (. Eric W. Weisstein et al., 'dd Perfect Numbers",) http://mathworld.wo1fram.com/OddPerfectNumber.html).

    http://pmes.utm.edu Chs Ca1dwell, Uniersity of Tennessee at Martin,

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    3.

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    3.1.

    q, : q. : , pz, ... Pv , :

    ( pz ... Pv) => q.

    , , q. :

    ! p =>q

    p =>q q p ; q

    ,

    . , , ' ( ), ( ).

    . 3.2.

    () :

    p => q, q => r, . .. , t => u, u =>, :

    p => q => r => ... => t => u => (1) p ( ). , p => ( ) p ( ). :

    I p ( =>) . . JR , :

    -3:::;; < 2 . :

    l2 + 3 l -6:: 2-3s; 2+ 3 - 7 < 2 + 3 < 7 => l2 + 3 1 < 7 .

    3.3. ""

    , . ,

    ' ' .4/6

  • , . :

    , u ( u ). t (t u).

    ... ' ,

    . :

    u, t, ... , , :

    u t ... . (2) (2) (1)

    , . :

    I ( ) . . , , e R.

    : + + = 1.

    : 2 + 2 + 2 ( + ) =2 + 1.

    . , :

    2 + 2 + 2+ 2 =2 + 1, 2 + 2 + 2 = 1 + 2 - 2,

    ( + ) 2 = ( 1- ) 2 , + = 1- , + + = 1 , .

    3.4. .

    I ( ) ( ) ' . , . :

  • ut ... r q p (4)

    ; , , , , , . p, :

    :::::> u :::::> t :::::> ... :::::> r :::::> q :::::> p (5) , , . :

    q .

    p , p :::::> q , q ( ).

    , p , q .

    p q. p , , p :::::> q .

    3.7.

    p:::::>q q:::::> p

    . : (:::::> q) (q :::::> p)

    ( ) , p :::::> q ,

    q :::::> p , p :::::> q . 3.8. : p q 1 . : p :::::> q q :::::> p . 2 . ( -

    ) : prt ... uq

    : q u ... r p.

    3 . p r q r .

    3.9. :

  • 0 ( 0 ), ( ), .

    - : Vx , () ( ), = p() ( ). a '\/ , ( ).

    - u , &xJ_ . , ' , , : Vx , (). . : !!!. , ( a ), , ( &xJ_ ) ( ).

    .

    ' , . 3. 10.

    ( ).

    . IR , : .J2x + 5 = + 1.

    . :

    .J2x + 5 =+ 1 + 1 {2+ 50

    2 + 5 = (+ 1 )2

    { + 1 { -1 2 2 + 5 = ( + 1) 2 + 5 = 2 + 2 + 1 {-1 { -1 = 2. 2=4 (= 2 = - 2)

    3.11. :

  • 3.12