「時計の世界の整数論」第2回プログラマのための数学勉強会 #maths4pg

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Transcript of 「時計の世界の整数論」第2回プログラマのための数学勉強会 #maths4pg

  • @tsujimotterhttp://tsujimotter.info

    2015 3 27

    2015/3/31 1

  • 2015/3/31 2

  • 2015/3/31 3

  • 2015/3/31 4

  • Disquisitiones Arithmeticae (D.A.)

    2015/3/31 5

    0 12

    34

    567

    8

    910

    11

    0

    1

    23

    4

    5

  • 2015/3/31 6

  • 2015/3/31 7

  • 2015/3/31 8

  • STEP 1:

    STEP 2:

    STEP 3: 2

    2015/3/31 9

  • 2015/3/31 10

  • STEP 1:

    STEP 2:

    STEP 3: 2

    2015/3/31 11

  • STEP 1:

    STEP 2:

    STEP 3: 2

    2015/3/31 12

  • , 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,

    2015/3/31 13

  • 01

    23

    4

    5

    6

    78

    9

    10

    11

    1213

    14

    15

    16

    1718

    19

    2015/3/31 14

  • 01

    23

    4

    5

    6

    78

    9

    10

    11

    1213

    14

    15

    16

    1718

    19

    2015/3/31 15

  • 01

    23

    4

    5

    6

    78

    9

    10

    11

    1213

    14

    15

    16

    1718

    19

    5 1 6 5 1 6 1 6 mod 5

    2015/3/31 16

  • 01

    23

    4

    5

    6

    78

    9

    10

    11

    1213

    14

    15

    16

    1718

    19

    1 6 11 16 mod 5

    2015/3/31 17

  • 01

    23

    4

    5 {0, 1, 2, 3, 4}

    2015/3/31 18

  • 0

    1

    23

    4

    0

    1

    23

    4

    0

    1

    23

    4

    1 + 3 4 (mod 5)2015/3/31 19

  • 0

    1

    23

    4

    0

    1

    23

    4

    0

    1

    23

    4

    3 + 4 2 (mod 5)

    (7)

    2015/3/31 20

  • 0

    1

    23

    4

    0

    1

    23

    4

    0

    1

    23

    4

    3 2 1 (mod 5)

    (6)

    2015/3/31 21

  • 01

    23

    4

    0

    1

    23

    4

    0

    1

    23

    4

    3 3 4 (mod 5)

    (9)

    2015/3/31 22

  • 01

    23

    4

    0

    1

    23

    4 2

    32 4 (mod 5)

    (9)

    2015/3/31 23

  • 01

    23

    4

    0

    1

    23

    4 2

    32 3 2 (mod 5)

    0

    1

    23

    4

    2015/3/31 24

  • 01

    23

    4

    0

    1

    23

    4

    4 3 2 (mod 5)

    0

    1

    23

    4

    (12)

    2015/3/31 25

  • 0

    1

    23

    4

    0

    1

    23

    4 3

    33 2 (mod 5)2015/3/31 26

  • n n1 n2 n3 n4

    3 3 4 2 1

    33 2 (mod 5)31 3 (mod 5)32 4 (mod 5)

    34 1 (mod 5)

    2015/3/31 27

  • n n1 n2 n3 n4

    1 1 1 1 1

    3 3 4 2 1

    2015/3/31 28

  • n n1 n2 n3 n4

    1 1 1 1 12 2 4 3 13 3 4 2 14 4 1 4 1

    2015/3/31 29

  • 01

    23

    4

    5

    n n1 n2 n3 n4

    1 1 1 1 12 2 4 3 13 3 4 2 14 4 1 4 1

    2015/3/31 30

    5

  • n n1 n2 n3 n4 n5 n6

    1 1 1 1 1 1 1

    2 2 4 1 2 4 1

    3 3 2 6 4 5 1

    4 4 2 1 4 2 1

    5 5 4 6 2 3 1

    6 6 1 6 1 6 1

    7

    2015/3/31 31

    01

    2

    34

    5

    6

    7

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    11

    2015/3/31 32

    0 12

    34

    567

    8

    910

    11

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10 n11 n12

    1 1 1 1 1 1 1 1 1 1 1 1 1

    2 2 4 8 3 6 12 11 9 5 10 7 1

    3 3 9 1 3 9 1 3 9 1 3 9 1

    4 4 3 12 9 10 1 4 3 12 9 10 1

    5 5 12 8 1 5 12 8 1 5 12 8 1

    6 6 10 8 9 2 12 7 3 5 4 11 1

    7 7 10 5 9 11 12 6 3 8 4 2 1

    8 8 12 5 1 8 12 5 1 8 12 5 1

    9 9 3 1 9 3 1 9 3 1 9 3 1

    10 10 9 12 3 4 1 10 9 12 3 4 1

    11 11 4 5 3 7 12 2 9 8 10 6 1

    12 12 1 12 1 12 1 12 1 12 1 12 1

    13

    2015/3/31 33

  • STEP 1:

    STEP 2:

    STEP 3: 2

    2015/3/31 34

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    1 1

    2015/3/31 35

  • 5 , 7 , 11 , 13 , 17 ,

    Webhttp://tsujimotter.info/maths4pg/2/

    2015/3/31 36

  • Go!!

    2015/3/31 37

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    2015/3/31 38

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    n10

    1p 0 (mod ) n

    1 1 mod 2015/3/31 39

    (1)

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    1,2,3, ,1 (mod )1, 2, 3, , 1

    1 10

    2015/3/31 40

    (2)

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    0 12

    3

    9

    456

    78

    10 21

    22

    23

    24

    25

    26

    27

    28

    29

    210

    0

    n = 2

    2015/3/31 41

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    2015/3/31 42

    (3) n2

    n2{1, 3, 4, 5, 9}

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    2015/3/31 43

    (3) n5

    1n {1, 3, 4, 5, 9}

    n2{1, 3, 4, 5, 9}

  • 0 12

    34

    567

    8

    9(-1)

    11

    0

    1

    23

    5

    (-1)

    4 -1 (mod 5)

    10 -1 (mod 11)

    2015/3/31 44

    p (-1) n2 p 4N + 1

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 (-1) 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 (-1) 5 8 4 2 17 7 5 2 3 (-1) 4 6 9 8 18 8 9 6 4 (-1) 3 2 5 7 19 9 4 3 5 1 9 4 3 5 1

    (-1) (-1) 1 (-1) 1 (-1) 1 (-1) 1 (-1) 1

    n2 (-1) 2015/3/31 45

    n5 n2

    (11 1) / 2 = 11 4N+3

    11

  • n n1 n2 n3 n4

    1 1 1 1 12 2 (-1) 3 13 3 (-1) 2 1

    (-1) (-1) 1 (-1) 1

    2015/3/31 46

    n2 (-1)

    n2

    (5 1) / 2 =

    5 4N+1

    5 N(5-1)/2

  • 5 = 22 + 1213 = 32 + 2217 = 42 + 1229 = 52 + 22

    7111923

    p p 4N + 1

    2015/3/31 47

    4N+3 4N+1

  • p (-1) n2 p 4N + 1

    p p 4N + 1

    2015/3/31 48

  • STEP 1:

    STEP 2:

    STEP 3: 2

    2015/3/31 49

  • n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    2015/3/31 50

  • n n1 n2 n3 n4

    1 1 1 1 12 2 4 3 13 3 4 2 14 4 1 4 1

    n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    2015/3/31 51

  • 5 11

    n n1 n2 n3 n4 n5 n6 n7 n8 n9 n10

    1 1 1 1 1 1 1 1 1 1 12 2 4 8 5 10 9 7 3 6 13 3 9 5 4 1 3 9 5 4 14 4 5 9 3 1 4 5 9 3 15 5 3 4 9 1 5 3 4 9 16 6 3 7 9 10 5 8 4 2 17 7 5 2 3 10 4 6 9 8 18 8 9 6 4 10 3 2 5 7 19 9 4 3 5 1 9 4 3 5 110 10 1 10 1 10 1 10 1 10 1

    1 11

    n n1 n2 n3 n4

    1 1 1 1 12 2 4 3 13 3 4 2 14 4 1 4 1

    5 n2 11 (1) 11 n2 5

    2015/3/31 52

  • 2015/3/31 53

    5 n2 11 (1) 11 n2 5

    5 n2 7 (2) 7 n2 5

  • 5 n2 11 (1) 11 n2 5

    5 n2 7 (2) 7 n2 5

    2015/3/31 54

    7 n2 11 (4) 11 n2 7

  • 5 n2 11 (1) 11 n2 5

    5 n2 7 (2) 7 n2 5

    2015/3/31 55

    4N+1

    4N+1

    4N+3

    4N+3

    7 n2 11 (4) 11 n2 74N+34N+3

    4n+3

  • 0

    1

    23

    4

    0 12

    34

    567

    8

    910

    (11)

    11 5 n2 5 11 n2

    2015/3/31 56

  • 01

    23

    4

    5

    (270329)

    270329 5 n2 5 270329(4) n2

    2015/3/31 57

  • 2015/3/31 58

  • STEP 1:

    STEP 2:

    STEP 3: 2

    2015/3/31 59

  • 2015/3/31 60

  • kamiyama2 http://www.nicovideo.jp/