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ɉɪɢɦɟɪ 3 ɇɚɨɫɧɨɜɚɧɢɢɉɪɢɦɟɪɚ
(
2346, 646
)
=2ή17 = 34.
ɇɟɞɨɫɬɚɬɨɤ ɷɬɨɝɨ ɫɩɨɫɨɛɚ ɫɨɫɬɨɢɬ ɜ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ
ɬɪɭɞɨɺɦɤɨɝɨɤɚɧɨɧɢɱɟɫɤɨɝɨɪɚɡɥɨɠɟɧɢɹ
ɛɜɵɱɢɫɥɟɧɢɟɇɈȾɩɨ ɚɥɝɨɪɢɬɦɭȿɜɤɥɢɞɚɉɨɫɥɟɞɧɢɣɧɟɧɭɥɟɜɨɣɨɫɬɚ
ɬɨɤɞɚɫɬɇɈȾ
ɉɪɢɦɟɪ 4 ȼɵɱɢɫɥɢɬɶɇɈȾɞɥɹɬɟɯɠɟɱɢɫɟɥɱɬɨɢɜɉɪɢɦɟɪɟ
2
2
0
1
1
68
34
3
1
238
408
Ⱦɟɥɢɬɟɥɢɜɫɹɤɢɣɪɚɡɫɬɚɧɨɜɹɬɫɹɞɟɥɢɦɵɦɢɚɨɫɬɚɬɤɢ– ɞɟɥɢɬɟɥɹɦɢȾɨɤɚ
ɡɵɜɚɟɬɫɹɱɬɨɚɥɝɨɪɢɬɦȿɜɤɥɢɞɚɜɫɟɝɞɚɤɨɧɟɱɟɧ
Ɉɫɬɚɬɤɢɜɵɞɟɥɟɧɵ ɠɢɪɧɵɦ ɲɪɢɮɬɨɦ ɉɨɫɥɟɞɧɢɣɧɟɧɭɥɟɜɨɣɨɫɬɚɬɨɤ ɪɚ
ɜɟɧɷɬɨɢɟɫɬɶɇɈȾ(2346, 646) ɫɪɫɉɪɢɦɟɪɨɦ
ɋɦɵɫɥ ɜɵɞɟɥɟɧɢɹ ɱɚɫɬɧɵɯ ɤɭɪɫɢɜɨɦ ɜɵɹɫɧɢɬɫɹ ɧɢɠɟ ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ
ɰɟɩɧɵɯɞɪɨɛɟɣ
ȼɵɱɢɫɥɟɧɢɟɇɈɄ
[
ɚ©ɲɤɨɥɶɧɵɣªɫɩɨɫɨɛɞɥɹɜɵɱɢɫɥɟɧɢɹ
]
ܽ, ܾ
ɫɨɫɬɚɜɢɬɶɤɚɧɨɧɢɱɟɫɤɨɟɪɚɡ
ɥɨɠɟɧɢɟɤɚɠɞɨɝɨɢɡɱɢɫɟɥɚɡɚɬɟɦɢɡɨɛɨɢɯɪɚɡɥɨɠɟɧɢɣɜɡɹɬɶɩɪɨɫɬɵɟɦɧɨɠɢ
ɬɟɥɢɩɪɢɫɭɬɫɬɜɭɸɳɢɟɯɨɬɹɛɵɜɨɞɧɨɦɪɚɡɥɨɠɟɧɢɢ– ɜɧɚɢɛɨɥɶɲɢɯ ɫɬɟɩɟɧɹɯ
ɉɪɢɦɟɪ 5 ɇɚɨɫɧɨɜɚɧɢɢɉɪɢɦɟɪɚ
[
2346, 646
]
=2ή3 ή17 ή19 ή23 = 44574.
Ɇɨɠɧɨɫɞɟɥɚɬɶɩɪɨɜɟɪɤɭɧɚɨɫɧɨɜɚɧɢɢ-ɝɨɫɜɨɣɫɬɜɚɇɈȾɢɇɈɄ
(
ܽ, ܾ)[ܽ, ܾ]= ܾܽ34 ή44574 = 2346 ή646.
ɛɜɵɱɢɫɥɟɧɢɟɇɈɄɱɟɪɟɡɩɨɫɪɟɞɫɬɜɨɇɈȾɫɩɨɦɨɳɶɸɚɥɝɨɪɢɬɦɚȿɜɤɥɢɞɚ
ɤɫɨɠɚɥɟɧɢɸɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɞɥɹɇɈɄɧɟɬ ɚɥɝɨɪɢɬɦɚɜɵɱɢɫɥɟɧɢɹɫɨɩɨɫɬɚ
ɜɢɦɨɝɨɫɚɥɝɨɪɢɬɦɨɦȿɜɤɥɢɞɚ
[
ܽ, ܾ
ܾܽ
]
=
(
ܽ, ܾ
.
)
Ⱦɨɩɨɥɧɢɬɟɥɶɧɵɟɩɪɢɦɟɪɵ
ɉɪɢɦɟɪ 6 (161, 3)=1 ɬɚɤɱɬɨɷɬɢɱɢɫɥɚɹɜɥɹɸɬɫɹɜɡɚɢɦɧɨɩɪɨɫɬɵɦɢ
53
2 1
2
2 2
0
2
ɉɪɢɦɟɪ (33, 13)=1 ɜɡɚɢɦɧɨɩɪɨɫɬɵɟɱɢɫɥɚ
2
1
1
6
6
0
ɉɪɢɦɟɪ 8 (17, 29)=1 ɜɡɚɢɦɧɨɩɪɨɫɬɵɟɱɢɫɥɚ
0 0
1
1
2
5 2
2
2
2 2
0
5
ɉɪɢɦɟɪ 9 (40, 23)=1 ɜɡɚɢɦɧɨɩɪɨɫɬɵɟɱɢɫɥɚ
1
1
2
5 1
5
5 5
0
ɉɪɢɦɟɪ (87, 55)=1 ɜɡɚɢɦɧɨɩɪɨɫɬɵɟɱɢɫɥɚ
55
55 1
55
1
1
2
5
5 1
5 4
1
4
0
6
5
23
9
32
ɉɪɢɦɟɪ (589, 343)=1 ɜɡɚɢɦɧɨɩɪɨɫɬɵɟɱɢɫɥɚ
1
1
2
52 1
52
1
6
3
2
3
0
ɉɪɢɦɟɪ (1403, 1058)=23
1
3
15
0
ɉɪɢɦɟɪ (1672, 1232)=88
1
2
1
352
88
4
0
52
45
345
23
440
246
ɉɪɢɦɟɪ (132, 21)=3
ɉɪɢɦɟɪ (8211, 135)=3
2
0
3
3
6
6
ɋɪɚɜɧɟɧɢɹɩɨɦɨɞɭɥɸ
ȼɵɫɤɚɡɵɜɚɧɢɟ©a ɫɪɚɜɧɢɦɨɫb ɩɨɦɨɞɭɥɸmª
ܽؠܾ(mod ݉
(
ɨɡɧɚɱɚɟɬɱɬɨɪɚɡɧɨɫɬɶ
ି
Ɍɨɝɞɚ
= ݇אԺ ܽ = ܾ + ݉݇
)
ܽെܾ
ɞɟɥɢɬɫɹɛɟɡɨɫɬɚɬɤɚɧɚm
(
ܽെܾ)ڭ݉
)
ɋɜɨɣɫɬɜɚɫɪɚɜɧɟɧɢɣ
ɚܽؠܾ(mod ݉)ؠ(ܽ+ ݉݇)(mod ݉)ؠ(ܾ+ ݉݇)(mod ݉), ݇אԺ . ɂɬɚɤ
ɤɥɸɛɨɣɱɚɫɬɢɫɪɚɜɧɟɧɢɹɦɨɠɧɨɰɟɥɨɟɱɢɫɥɨɪɚɡɩɪɢɛɚɜɢɬɶɦɨɞɭɥɶɫɪɚɜɧɟɧɢɹ
ɛȿɫɥɢܽؠܾ(mod ݉) ɢܿؠ݀(mod ݉
)
ɬɨ
ܽήܿؠܾή݀(mod ݉
)
ɬ ɟɫɪɚɜɧɟɧɢɹɩɨɨɞɧɨɦɭ ɦɨɞɭɥɸɦɨɠɧɨɩɟɪɟɦɧɨɠɚɬɶ
ɋɥɟɞɫɬɜɢɟɨɬɫɸɞɚ
ؠܾ
ɜȿɫɥɢ ܽؠܾ(mod ݉)ɬɨܽ
ɉɪɢɦɟɪ 21 ؠ 7
(
mod 2
)
ɬ ɤɪɚɡɧɨɫɬɶ21 െ7=14ڭ 2.
(
mod ݉)݊אԺ
ɐɟɩɧɵɟɧɟɩɪɟɪɵɜɧɵɟɞɪɨɛɢ
ɇɚɱɧɺɦɫɩɪɢɦɟɪɚɩɪɟɨɛɪɚɡɨɜɚɧɢɹɩɪɨɫɬɨɣɞɪɨɛɢ
ɉɪɢɦɟɪ
25
7
=3+
4
=3+
7
1
=3+
ସ
1
1+
=3+
ଷ
ସ
1
1+
=3+
ଵ
ర
య
1+
1
=[3; 1, 1, 3].
ଵ
భ
ଵା
య
ɉɨɥɭɱɟɧɚɰɟɩɧɚɹɞɪɨɛɶɡɚɩɢɫɚɧɧɚɹɫɧɚɱɚɥɚɜɪɚɡɜɺɪɧɭɬɨɦɚɡɚɬɟɦ– ɜɫɨ
ɤɪɚɳɺɧɧɨɦɜɢɞɟ
ɗɬɨɬɩɪɢɦɟɪɢɥɥɸɫɬɪɢɪɭɟɬ ɬɟɨɪ ɟɦɭ ɨɫɭɳɟɫɬɜɨɜɚɧɢɢɟɞɢɧɫɬɜɟɧɧɨɣɤɨ
ɧɟɱɧɨɣ ɰɟɩɧɨɣɞɪɨɛɢɜɤɨɬɨɪɭɸɪɚɫɤɥɚɞɵɜɚɟɬɫɹɥɸɛɨɟɪɚɰɢɨɧɚɥɶɧɨɟ ɱɢɫɥɨ
ɉɪɢɦɟɪ Ɋɚɡɥɨɠɢɦ ɢɪɪɚɰɢɨɧɚɥɶɧɨɟ ɱɢɫɥɨξ2
ɜ ɰɟɩɧɭɸ ɞɪɨɛɶ
ɇɚɱɧɺɦɫɥɟɝɤɨɩɪɨɜɟɪɹɟɦɨɝɨɬɨɠɞɟɫɬɜɚ
ξ
=1+
2
ଵ
ଵାξଶ
ɉɨɞɫɬɚɜɥɹɹɜɡɧɚɦɟɧɚɬɟɥɟɜɦɟɫɬɨξ2
ɟɝɨ ɜɵɪɚɠɟɧɢɟɢɡ ɬɨɠɞɟɫɬɜɚ ɩɨɥɭ
ɱɢɦ
ξ
=1+
2
ଵାቀଵା
ଵ
భ
భశξమ
=1+
ቁ
ଶା
ଵ
భ
భశξమ
=1+
ଶା
భశ൬భశ
ଵ
=1+
భ
భ
൰
భశξమ
ଶା
ଵ
= ڮ
భ
భ
మశ
భశξమ
ത
]
ɢɬɨɝɨξ2
=[1; 2,2, 2, …]=[1; 2
ɗɬɨɬɩɪɢɦɟɪɢɥɥɸɫɬɪɢɪɭɟɬ ɬ ɟɨɪɟ ɦɭ ɨɫɭɳɟɫɬɜɨɜɚɧɢɢɟɞɢɧɫɬɜɟɧɧɨɣɛɟɫ
ɤɨɧɟɱɧɨɣɩɟɪɢɨɞɢɱɟɫɤɨɣ ɰɟɩɧɨɣɞɪɨɛɢɜɤɨɬɨɪɭɸɪɚɫɤɥɚɞɵɜɚɟɬɫɹɥɸɛɚɹɤɜɚɞ
ɪɚɬɢɱɧɚɹ ɢɪɪɚɰɢɨɧɚɥɶɧɨɫɬɶ ɬɢɩɚξ2
,ξ3,ξ5,…
ɉɪɢɦɟɪ ɂɡɜɟɫɬɧɵɟɦɚɬɟɦɚɬɢɱɟɫɤɢɟɤɨɧɫɬɚɧɬɵ– ɱɢɫɥɚɟ ɢߨ – ɪɚɫɤɥɚ
ത
ɞɵɜɚɸɬɫɹɜɫɥɟɞɭɸɳɢɟɛɟɫɤɨɧɟɱɧɵɟɧɟɩɟɪɢɨɞɢɱɟɫɤɢɟ ɰɟɩɧɵɟɞɪɨɛɢ
ஶ
]
݁ =[2; 1,2݊,1
ߨ =
[
3; 7,15,1,292,1,1,1,2,1,3,1,14,…]
[
2; 1, 2, 1, 1, 4, 1, 1, 6, 1, …],
=
ୀଵ
Ⱦɥɹɨɫɧɨɜɚɧɢɹɧɚɬɭɪɚɥɶɧɵɯɥɨɝɚɪɢɮɦɨɜɰɟɩɧɚɹɞɪɨɛɶɛɭɞɭɱɢɧɟɩɟɪɢɨɞɢ
ɱɟɫɤɨɣɢɦɟɟɬɩɪɨɡɪɚɱɧɨɟɫɬɪɨɟɧɢɟɬɨɝɞɚɤɚɤɞɥɹɱɢɫɥɚߨ ɧɢɤɚɤɚɹɡɚɤɨɧɨɦɟɪ
ɧɨɫɬɶɫɥɟɞɨɜɚɧɢɹɱɢɫɟɥɧɟɢɡɜɟɫɬɧɚ
Ɉɛɪɵɜɚɹ ɰɟɩɧɭɸ ɞɪɨɛɶ ɧɚ ɧɟɤɨɬɨɪɨɦ ɡɜɟɧɟ ɩɨɥɭɱɢɦ ɩɨɞɯɨɞɹɳɭɸ
ɞɪɨɛɶ– ɩɪɨɫɬɭɸɞɪɨɛɶɹɜɥɹɸɳɭɸɫɹɧɚɢɥɭɱɲɢɦɪɚɰɢɨɧɚɥɶɧɵɦɩɪɢɛɥɢɠɟɧɢɟɦ
ɢɫɯɨɞɧɨɣɰɟɩɧɨɣɞɪɨɛɢ
ɉɪɢɦɟɪ 20 ɉɨɥɭɱɢɦɩɟɪɜɵɟɩɨɞɯɨɞɹɳɢɟɞɪɨɛɢɞɥɹɱɢɫɥɚߨ
ɇɭɥɟɜɚɹ ɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶ– ɰɟɥɚɹɱɚɫɬɶɱɢɫɥɚ
ܲ
ܳ
=3=
3
;
1
ɩɟɪɜɚɹ ɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶɩɨɥɭɱɚɟɬɫɹɩɪɢɭɱɺɬɟɨɞɧɨɝɨɞɪɨɛɧɨɝɨɡɜɟɧɚ
ܲ
ଵ
ܳ
ଵ
=3+
1
22
=
7
ൎ 3,143
7
ɚɪɯɢɦɟɞɨɜɨɱɢɫɥɨ
ɜɬɨɪɚɹ ɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶ
ܲ
ଶ
ܳ
ଶ
=3+
1
7+
1
15
333
=
106
ൎ 3,14151;
ɬɪɟɬɶɹ ɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶ
ܲ
ܳ
ଷ
=3+
ଷ
7+
1
1
15 +
355
=
ൎ 3,1415929;
113
1
1
ɱɟɬɜɺɪɬɚɹ ɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶ
ܲ
ସ
ܳ
=3+
ସ
7+
15 +
1
1
1+
1
292
1
103993
=
33102
ൎ 3,1415926530
ɢɬ ɞ
Ⱦɥɹɫɪɚɜɧɟɧɢɹɩɪɢɜɟɞɺɦɞɟɫɹɬɢɱɧɨɟɩɪɟɞɫɬɚɜɥɟɧɢɟɱɢɫɥɚߨ
ߨ = 3,14159265359 …
ɉɪɢ ɭɱɺɬɟ ɤɚɠɞɨɝɨɞɨɛɚɜɨɱɧɨɝɨ ɡɜɟɧɚ ɰɟɩɧɨɣ ɞɪɨɛɢ ɤɨɥɢɱɟɫɬɜɨɜɟɪɧɵɯ
ɞɟɫɹɬɢɱɧɵɯɡɧɚɤɨɜɩɪɢɪɚɫɬɚɟɬɜɫɪɟɞɧɟɦɧɚɞɜɚ
ɉɨɫɤɨɥɶɤɭɤɚɠɞɚɹɩɨɫɥɟɞɭɸɳɚɹɩɨɞɯɨɞɹɳɚɹɞɪɨɛɶɜɵɱɢɫɥɹɟɬɫɹɜɫɺɛɨɥɟɟ
ɬɪɭɞɨɺɦɤɨɩɪɹɦɨɟɜɵɱɢɫɥɟɧɢɟɩɨɞɯɨɞɹɳɢɯɞɪɨɛɟɣɧɟɭɞɨɛɧɨɨɛɵɱɧɨɩɨɥɶɡɭ
ɸɬɫɹɫɥɟɞɭɸɳɢɦɩɪɚɜɢɥɨɦ
ɩɟɪ ɜɵɟ ɞɜ ɟɩɨɞɯɨɞɹɳɢɯɞɪɨɛɢɞɥɹɰɟɩɧɨɣɞɪɨɛɢ
[
ݍ
; ݍଵ, ݍଶ, ݍ
,…]
ଷ
ɜɵɱɢɫɥɹɸɬɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ
ܲ
= ݍ
ܳ
ܲ
ଵ
= ݍ
+
ܳ
ଵ
ݍ
=
,
1
1
ݍ
+1
ݍଵ
=
ݍ
ଵ
,
ݍ
ଵ
ɚɞɥɹɧɚɯɨɠɞɟɧɢɹɩɨɫɥɟɞɭɸɳɢɯɩɨɞɯɨɞɹɳɢɯɞɪɨɛɟɣ ɩɪɢɦɟɧɹɸɬɪ ɟɤɭɪɪ ɟɧɬ
ɧɭɸɮɨɪɦɭɥɭ
ܲ
ݍ
=
ܳ
ݍܳ
ܲିଵ
ିଵ
+ ܲ
+ ܳ
ିଶ
, ݇ =2,3,…
ିଶ
Ɂɞɟɫɶ ݍ
– ɱɚɫɬɧɵɟ ɤɨɬɨɪɵɟ ɜɨɡɧɢɤɚɸɬ ɩɪɢ ɪɟɚɥɢɡɚɰɢɢ ɚɥɝɨɪɢɬɦɚ ȿɜɤɥɢɞɚ
ɨɧɢɩɨɦɟɱɟɧɵɤɭɪɫɢɜɨɦɜɜɵɲɟɥɟɠɚɳɢɯɩɪɢɦɟɪɚɯɌɚɤɢɦɨɛɪɚɡɨɦɚɥɝɨɪɢɬɦ
ȿɜɤɥɢɞɚ ɩɪɢɦɟɧɹɟɬɫɹ ɧɟ ɬɨɥɶɤɨ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɇɈȾ ɧɨ ɢ ɞɥɹ ɪɚɡɥɨɠɟɧɢɹ
ɩɪɨɫɬɨɣɞɪɨɛɢɜɰɟɩɧɭɸ
ସ
ɉɪɢɦɟɪ Ɋɚɡɥɨɠɢɬɶ ɩɪɨɫɬɭɸ ɞɪɨɛɶ
ɜɰɟɩɧɭɸɢɧɚɣɬɢɞɥɹɧɟɺɜɫɟ
ଶଷ
ɩɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ʑ ʞ˓ˋˏˈ˓ˈ 9 ˄˞ˎˑ ˅˞˒ˑˎːˈːˑ ˇˈˎˈːˋˈ 40 ː˃ 23 ˒ˑ ˃ˎˆˑ˓ˋ˕ˏ˖ ʔ˅ˍˎˋˇ˃
1
5
5
5
5 5
0
2
1
1
Ɋɚɡɥɨɠɟɧɢɟ ɜɰɟɩɧɭɸɞɪɨɛɶɢɦɟɟɬɜɢɞ
1+
1
2+
1
1
1+
.
1
5
40
23
[
=
ݍ; ݍଵ, ݍଶ, ݍଷ, ݍ
]=[
1; 1, 2, 1, 5]=1+
ସ
ɉɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ɧɭɥɟɜɚɹ
ܲ
ݍ
=
ܳ
1
=
,
1
1
ɩɟɪɜɚɹ
ܲ
ଵ
= ݍ
ܳ
ଵ
1
+
ݍ
ଵ
=1+
1
2
=
;
1
1
ɞɚɥɟɟ– ɩɨɪɟɤɭɪɪɟɧɬɧɨɣɮɨɪɦɭɥɟ
ɜɬɨɪɚɹ
ܲ
ଶ
ܳ
ଶ
ݍ
ଶܲଵ
=
ݍଶܳଵ+ ܳ
+ ܲ
=
2 ή2+1
2 ή1+1
5
=
,
3
ɬɪɟɬɶɹ
ܲ
ଷ
ܳ
ଷ
ݍ
ଷܲଶ
=
ݍଷܳଶ+ ܳ
+ ܲ
ଵ
=
ଵ
1 ή5+2
1 ή3+1
7
=
,
4
ɱɟɬɜɺɪɬɚɹ
ܲ
ସ
ܳ
=
ସ
ݍ
+ ܲ
ସܲଷ
ݍସܳଷ+ ܳ
ଶ
=
ଶ
5 ή7+5
5 ή4+3
40
=
.
23
ɉɨɥɭɱɟɧɚɢɫɯɨɞɧɚɹɞɪɨɛɶ
ଵ
ɉɪɢɦɟɪ 22 Ɋɚɡɥɨɠɢɬɶ ɩɪɨɫɬɭɸ ɞɪɨɛɶ
ɜɰɟɩɧɭɸɢɧɚɣɬɢɞɥɹɧɟɺɜɫɟ
ଶଽ
ɩɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ʑ ʞ˓ˋˏˈ˓ˈ 8 ˄˞ˎˑ ˅˞˒ˑˎːˈːˑ ˇˈˎˈːˋˈ 17 ː˃ 29 ˒ˑ ˃ˎˆˑ˓ˋ˕ˏ˖ ʔ˅ˍˎˋˇ˃
0 0
1
1
2
5 2
2
2
2 2
0
5
Ɋɚɡɥɨɠɟɧɢɟ ɜɰɟɩɧɭɸɞɪɨɛɶɢɦɟɟɬɜɢɞ
17
[
=
ݍ; ݍଵ, ݍଶ, ݍଷ, ݍସ, ݍ
29
]=[
ହ
ɉɨɞɯɨɞɹɳɢɟɞɪɨɛɢ
ɧɭɥɟɜɚɹ
0; 1,1, 2, 2, 2]=0+
ݍ
ܲ
=
ܳ
0
=
,
1
1
1+
1+
1
1
2+
1
1
2+
.
1
2
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