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ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɨ ɦɧɨɝɢɯ ɩɟɪɟɛɨɪɧɵɯ ɡɚɞɚɱɚɯ ɬɪɟɛɭɟɬɫɹ ɪɚɫɫɦɨɬɪɟɬɶ ɜɫɟ
ɩɨɞɦɧɨɠɟɫɬɜɚ ɧɟɤɨɬɨɪɨɝɨ ɦɧɨɠɟɫɬɜɚ ɢ ɧɚɣɬɢ ɫɪɟɞɢ ɧɢɯ ɬɨ, ɤɨɬɨɪɨɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɡɚɞɚɧɧɨɦɭ ɭɫɥɨɜɢɸ. ɉɪɢ ɷɬɨɦ ɩɪɨɜɟɪɤɚ ɭɫɥɨɜɢɹ ɱɚɫɬɨ ɦɨɠɟɬ ɛɵɬɶ
ɜɟɫɶɦɚ ɬɪɭɞɨɟɦɤɨɣ ɢ ɡɚɜɢɫɟɬɶ ɨɬ ɫɨɫɬɚɜɚ ɷɥɟɦɟɧɬɨɜ ɨɱɟɪɟɞɧɨɝɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɩɨɞɦɧɨɠɟɫɬɜɚ. ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɨɱɟɪɟɞɧɨɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɨɬɥɢɱɚɟɬɫɹ ɩɨ ɫɨɫɬɚɜɭ
ɷɥɟɦɟɧɬɨɜ ɨɬ ɩɪɟɞɵɞɭɳɟɝɨ,
ɬɨ ɢɧɨɝɞɚ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɪɟɡɭɥɶɬɚɬɚɦɢ ɨɰɟɧɤɢ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟ
ɪɚɫɫɦɚɬɪɢɜɚɥɢɫɶ ɧɚ ɩɪɟɞɵɞɭɳɟɦ ɲɚɝɟ ɩɟɪɟɛɨɪɚ. ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɩɟɪɟɛɢɪɚɬɶ ɩɨɞɦɧɨɠɟɫɬɜɚ ɜ ɩɨɞɯɨɞɹɳɟɦ ɩɨɪɹɞɤɟ, ɦɨɠɧɨ ɡɧɚɱɢɬɟɥɶɧɨ ɭɫɤɨɪɢɬɶ
ɩɨɢɫɤ (ɪɚɛɨɬɭ ɩɟɪɟɛɨɪɧɨɝɨ ɦɟɯɚɧɢɡɦɚ). Ɋɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɢɣ ɚɥɝɨɪɢɬɦ:
Ⱥɥɝɨɪɢɬɦ 8 (ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɜɫɟɯ ɩɨɞɦɧɨɠɟɫɬɜ ɦɧɨɠɟɫɬɜɚ).
n=+prompt('ȼɜɟɞɢ ɦɨɳɧɨɫɬɶ ɦɧɨɠɟɫɬɜɚ n','2'); t='';
for(i=0; i<n; i++){u=t.split('\n'); t=''; for(j=0; j<u.length; j++)t+=u[j]+'0\n';
for(j=u.length-1; j>=0; j--)t+=u[j]+'1\n'; t=t.substr(0,t.length-1)}t
= 0 ɢɫɤɨɦɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɤɨɞɨɜ
Ɉɛɨɫɧɭɟɦ ɷɬɨɬ ɚɥɝɨɪɢɬɦ.
Ⱦɥɹ n
ɟɫɬɶ «0», ɞɥɹ n = 1 – «0, 1». ɉɭɫɬɶ ɢɡɜɟɫɬɧɚ ɢɫɤɨɦɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɤɨɞɨɜ
», «ɋ1», …, «ɋm», ɞɥɹ m = 2k – 1 ɢ k = n. Ɍɨɝɞɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ «ɋ00»,
«ɋ
0
0», …, «ɋm0», «ɋm1», «ɋ
«ɋ
1
ɫɬɶɸ ɞɥɹ k = n + 1. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ «ɋ
1», …, «ɋ01» ɢɦɟɟɬɫɹ 2
«ɋ
m
1», …, «ɋ01» ɛɭɞɟɬ ɢɫɤɨɦɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ-
m–1
k+1
ɤɨɞɨɜ, ɨɧɢ ɜɫɟ ɪɚɡɥɢɱɧɵ ɢ ɫɨɫɟɞɧɢɟ ɷɥɟɦɟɧɬɵ
0», …, «ɋm0»,
0
ɪɚɡɥɢɱɚɸɬɫɹ ɪɨɜɧɨ ɜ ɨɞɧɨɦ ɪɚɡɪɹɞɟ ɩɨ ɩɨɫɬɪɨɟɧɢɸ.
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɨɜ 7 ɢ 8 ɞɥɹ n = 4
i Ʉɨɞɵ
(7)
0
0000
1
1000
2
0100
3
1100
4
0010
5
1010
6
0110
7
1110
8
0001
9
1001
10
0101
11
1101
12
0011
13
1011
14
0111
15
1111
Ʉɨɞɵ C
(8)
0000
1000
1100
0100
0110
1110
1010
0010
0011
1011
1111
0111
0101
1101
1001
0001
21
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɧɚɦ ɢɡ ɷɬɢɯ ɩɨɞɦɧɨɠɟɫɬɜ ɭɞɨɛɧɨ ɨɛɪɚɛɚɬɵɜɚɬɶ
ɬɨɥɶɤɨ k-ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ k = 1, 2, … , n. ɋɥɟɞɭɸɳɢɣ ɚɥɝɨɪɢɬɦ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɚɬɶ ɢɯ ɫɩɢɫɤɢ k-ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜ k = 1, 2, … , n:
Ⱥɥɝɨɪɢɬɦ 9 (ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɜɫɟɯ ɩɨɞɦɧɨɠɟɫɬɜ ɦɧɨɠɟɫɬɜɚ).
n=+prompt('ȼɜɟɞɢ ɦɨɳɧɨɫɬɶ ɦɧɨɠɟɫɬɜɚ n','2'); t='';
for(i=0; i<n; i++){u=t.split(','); t=''; for(j=0; j<u.length; j++)t+=u[j]+'0,';
for(j=u.length-1; j>=0; j--)t+=u[j]+'1,'; t=t.substr(0,t.length-1)}
u=t.split(','); for(i=0; i<=n; i++){w[i]=0; v[i]=''}
for(i=0; i<u.length; i++){j=u[i].split('1').length-1; w[j]++; v[j]+=u[i]+' '};
for(i=0; i<=n; i++)t+='\nɱɢɫɥɨ '+i+'-ɷɥɟɦɟɧɬɧɵɯ '+w[i]+': '+v[i];
'ɉɨɞɦɧɨɠɟɫɬɜɚ:\n'+t
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 9 ɞɥɹ n = 4
ɉɨɞɦɧɨɠɟɫɬɜɚ:
0000,1000,1100,0100,0110,1110,1010,0010,0011,1011,1111,0111,0101,1101,
1001,0001
ɱɢɫɥɨ 0-ɷɥɟɦɟɧɬɧɵɯ 1: 0000
ɱɢɫɥɨ 1-ɷɥɟɦɟɧɬɧɵɯ 4: 1000 0100 0010 0001
ɱɢɫɥɨ 2-ɷɥɟɦɟɧɬɧɵɯ 6: 1100 0110 1010 0011 0101 1001
ɱɢɫɥɨ 3-ɷɥɟɦɟɧɬɧɵɯ 4: 1110 1011 0111 1101
ɱɢɫɥɨ 4-ɷɥɟɦɟɧɬɧɵɯ 1: 1111
ɇɚɩɨɦɧɢɦ, ɱɬɨ ɱɢɫɥɚ, ɤɨɬɨɪɵɟ ɧɚɯɨɞɢɬ ɷɬɨɬ ɚɥɝɨɪɢɬɦ, ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɵ, ɧɚɡɵɜɚɸɬɫɹ ɱɢɫɥɚɦɢ ɫɨɱɟɬɚɧɢɣ (ɱɢɫɥɨ k-ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜ nɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨɦ ɫɨɱɟɬɚɧɢɣ ɢɡ n ɷɥɟɦɟɧɬɨɜ ɩɨ k)
ɢ ɨɛɨɡɧɚɱɚɸɬɫɹ
ɩɪɢ m > n. ɗɬɢ ɱɢɫɥɚ ɫɜɹɡɚɧɵ ɪɟɤɭɪɪɟɧɬɧɵɦ ɫɨɨɬɧɨɲɟɧɢɟɦ:
k
C
. Ɉɱɟɜɢɞɧɨ, ɱɬɨ
n
n
10
CC
, ɩɪɢ n = 0, 1, 2, …, ɢ
nn
k
k
k
1
CCC
. (4)
n
n
n
1
1
m
0
C
n
ȼ ɫɚɦɨɦ ɞɟɥɟ, ɩɪɢ n > 0 ɢ 0 < k < n ɜɫɟ k-ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ
ɦɧɨɠɟɫɬɜɚ {1, 2, ..., n} ɦɨɠɧɨ ɪɚɡɛɢɬɶ ɧɚ ɞɜɚ ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɪɚɜɧɨɦɨɳɧɵɯ ɤɥɚɫɫɚ: k-ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜɚ {1, 2, ..., n – 1} ɢ (k – 1)ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜɚ {1, 2, ..., n – 1}, ɨɛɴɟɞɢɧɟɧɧɵɟ
ɫ ɨɞɧɨɷɥɟɦɟɧɬɧɵɦ ɦɧɨɠɟɫɬɜɨɦ {n}. ɉɨɷɬɨɦɭ ɱɢɫɥɨ ɜɫɟɯ ɬɚɤɢɯ ɩɨɞɦɧɨɠɟɫɬɜ
ɪɚɜɧɨ ɫɭɦɦɟ ɷɥɟɦɟɧɬɨɜ ɤɚɠɞɨɝɨ ɢɡ ɷɬɢɯ ɤɥɚɫɫɨɜ.
Ɍɨɠɞɟɫɬɜɨ (4) ɡɚɩɢɫɵɜɚɟɬɫɹ
ɜ ɜɢɞɟ ɬɪɟɭɝɨɥɶɧɢɤɚ, ɢɡɜɟɫɬɧɨɝɨ ɤɚɤ ɬɪɟ-
ɭɝɨɥɶɧɢɤ ɉɚɫɤɚɥɹ:
1
1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
…
,
22
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɂɚɦɟɬɢɦ, ɱɬɨ ɱɢɫɥɨ (n – k)-ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜ ɪɚɜɧɨ ɱɢɫɥɭ
k-ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜ, ɬɚɤ ɤɚɤ ɤɚɠɞɨɦɭ k-ɷɥɟɦɟɧɬɧɨɦɭ ɩɨɞɦɧɨɠɟɫɬɜɭ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ (n – k)-ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜɨ (ɟɝɨ ɞɨɩɨɥɧɟɧɢɟ),
ɢ ɧɚɨɛɨɪɨɬ. Ɍɨ ɟɫɬɶ ɜɟɪɧɚ ɮɨɪɦɭɥɚ:
kn
k
CC
n
n
Ⱥɥɝɨɪɢɬɦ 9 ɩɨɥɭɱɟɧɢɹ k-ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜ, ɬɚɤ ɤɚɤ ɞɥɹ ɝɟɧɟɪɚɰɢɢ ɷɬɢɯ ɩɨɞɦɧɨɠɟɫɬɜ ɩɪɢɯɨɞɢɬɫɹ ɫɧɚɱɚɥɚ ɧɚɯɨɞɢɬɶ ɜɫɟ ɩɨɞɦɧɨɠɟɫɬɜɚ. Ɉɞɧɚɤɨ ɮɨɪɦɭɥɚ (4) ɢ ɟɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɩɪɢɜɨɞɹɬ ɧɚɫ ɤ ɫɥɟɞɭɸɳɟɦɭ ɪɟɤɭɪɫɢɜɧɨɦɭ ɚɥɝɨɪɢɬɦɭ.
ɗɬɨɬ ɚɥɝɨɪɢɬɦ ɝɟɧɟɪɢɪɭɟɬ ɜɫɟ k-ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨ ɤɚɠɞɨɟ ɩɨɫɥɟɞɭɸɳɟɟ ɩɨɞɦɧɨɠɟɫɬɜɨ ɨɛɪɚɡɭɟɬɫɹ ɢɡ ɩɪɟɞɵɞɭɳɟɝɨ
ɭɞɚɥɟɧɢɟɦ ɨɞɧɨɝɨ ɷɥɟɦɟɧɬɚ
ɢ ɞɨɛɚɜɥɟɧɢɟɦ ɞɪɭɝɨɝɨ. Ɉɛɨɡɧɚɱɢɦ ɫ ɷɬɨɣ ɰɟɥɶɸ
ɱɟɪɟɡ G(n, k) ɫɩɢɫɨɤ, ɫɨɞɟɪɠɚɳɢɣ ɜɫɟ k-ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜɚ
{1, 2, ..., n}, ɜ ɤɨɬɨɪɨɦ ɩɟɪɜɵɦ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɹɜɥɹɟɬɫɹ {1, 2, ..., k},
ɚ ɩɨɫɥɟɞɧɢɦ – {1, 2, ..., k – 1, n}, ɢ ɤɚɠɞɨɟ ɫɥɟɞɭɸɳɟɟ ɩɨɞɦɧɨɠɟɫɬɜɨ ɨɛɪɚɡɭɟɬɫɹ ɢɡ ɩɪɟɞɵɞɭɳɟɝɨ ɡɚɦɟɧɨɣ ɧɟɤɨɬɨɪɨɝɨ ɷɥɟɦɟɧɬɚ ɞɪɭɝɢɦ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ G(n – 1, k) ɢ G
(n – 1, k – 1) ɭɠɟ ɩɨɫɬɪɨɟɧɵ, ɬɨ G(n, k) ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
G(n, k) = G(n – 1, k), G
(n – 1, k – 1) {n},
1
(n – 1, k) {n} ɨɛɨɡɧɚɱɚɟɬ ɫɩɢɫɨɤ, ɨɛɪɚɡɨɜɚɧɧɵɣ ɢɡ G(n – 1, k – 1) ɢɡ-
ɝɞɟ G
1
ɦɟɧɟɧɢɟɦ ɷɥɟɦɟɧɬɨɜ ɫɩɢɫɤɚ ɧɚ ɨɛɪɚɬɧɵɣ ɢ ɩɨɫɥɟɞɭɸɳɢɦ ɞɨɛɚɜɥɟɧɢɟɦ ɷɥɟɦɟɧɬɚ n ɤ ɤɚɠɞɨɦɭ ɩɨɞɦɧɨɠɟɫɬɜɭ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, G(n – 1, k) ɫɨɞɟɪɠɢɬ ɜɫɟ
k-ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜɚ {1, 2, ..., n}, ɧɟ ɫɨɞɟɪɠɚɳɢɟ n,
G
(n – 1, k – 1) {n} – ɜɫɟ k-ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ, ɫɨɞɟɪɠɚɳɢɟ n, ɩɪɢ-
1
ɱɟɦ ɩɨɫɥɟɞɧɢɦ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɜ ɫɩɢɫɤɟ G(n – 1, k) ɹɜɥɹɟɬɫɹ {1, 2, ..., k – 1, n – 1},
ɚ ɩɟɪɜɵɦ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɜ ɫɩɢɫɤɟ G
(n – 1, k – 1) {n} ɹɜɥɹɟɬɫɹ {1, 2, ...,
1
k – 1, n}. Ɉɞɧɚɤɨ ɡɚɦɟɬɢɦ, ɱɬɨ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ G(n + 1, k) ɩɨɬɪɟɛɭɟɬɫɹ ɫɬɪɨɢɬɶ
G(n, k – 1).
ȼ ɩɪɨɬɨɤɨɥɟ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 9 ɫɩɢɫɨɤ 2-ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜ ɦɧɨɠɟɫɬɜɚ {1, 2, 3, 4} ɩɨɥɭɱɚɟɬɫɹ ɬɚɤɢɦ: {1, 2}, {2, 3} {1, 3} {3, 4}
{2, 4} {1, 4} (1100, 0110, 1010, 0011, 0101, 1001). ȼɨɡɧɢɤɚɟɬ ɜɨɩɪɨɫ: ɧɟɥɶɡɹ ɥɢ
ɤɚɤ-ɧɢɛɭɞɶ ɭɩɨɪɹɞɨɱɢɬɶ ɷɬɨɬ ɫɩɢɫɨɤ? ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɜɨɩɪɨɫɨɦ ɪɚɫɫɦɨɬɪɢɦ
ɫɥɟɞɭɸɳɢɣ ɚɥɝɨɪɢɬɦ.
Ȼɭɞɟɦ ɜɵɛɪɚɬɶ
k-ɷɥɟɦɟɧɬɧɵɟ ɩɨɞɦɧɨɠɟɫɬɜɚ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ {1, 2, ..., n}. Ɍɨɝɞɚ ɤɚɠɞɨɦɭ ɬɚɤɨɦɭ ɩɨɞɦɧɨɠɟɫɬɜɭ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɨɡɪɚɫɬɚɸɳɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɞɥɢɧɵ k ɫ ɷɥɟɦɟɧɬɚɦɢ ɢɡ
{1, 2, ..., n}. ɇɚɩɪɢɦɟɪ, ɩɨɞɦɧɨɠɟɫɬɜɭ {4, 1, 2} ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɫɥɟɞɨɜɚ-
ɬɟɥɶɧɨɫɬɶ <1, 2, 4>. Ɇɵ ɛɭɞɟɦ ɝɟɧɟɪɢɪɨɜɚɬɶ ɬɚɤɢɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
ɜ ɚɥɮɚɜɢɬɧɨɦ (ɥɟɤɫɢɤɨɝɪɚɮɢɱɟɫɤɨɦ) ɩɨɪɹɞɤɟ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɬɚɤɨɦ
23
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɩɨɪɹɞɤɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɸ, ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɥɟɞɭɸɳɟɣ ɡɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɸ <a
, …, ak>, ɹɜɥɹɟɬɫɹ
1
, …, bk> = <a1, …, a
<b
1
, ap +1, ap +2, …, ap +k – p + 1>,
p–1
ɝɞɟ ɪ = max{i: a
ɧɨ ɫɥɟɞɭɸɳɚɹ ɡɚ <b
< n – k + 1}. Ȼɨɥɟɟ ɬɨɝɨ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ, ɧɟɩɨɫɪɟɞɫɬɜɟɧ-
i
, …, bk>, ɹɜɥɹɟɬɫɹ
1
, …, ɫk> = <b1, …, b
<ɫ
1
, bq +1, bq +2, …, bq +k – q + 1>,
q-1
ɝɞɟ
,1
q
®
,
¯
nbɟɫɥɢp
k
nbɟɫɥɢk
,
k
ɟɫɬɟɫɬɜɟɧɧɨ, ɱɬɨ ɡɞɟɫɶ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ <a
, …, ak> ɢ <b1, …, bk> ɨɬɥɢɱɚ-
1
ɸɬɫɹ ɨɬ ɩɨɫɥɟɞɧɟɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ <n – k + 1, …, n> ɜ ɧɚɲɟɦ ɩɨɪɹɞɤɟ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɢɣ ɩɪɨɫɬɨɣ ɚɥɝɨɪɢɬɦ:
Ⱥɥɝɨɪɢɬɦ 10 (ɦɨɞɟɥɢɪɨɜɚɧɢɟ k-ɷɥɟɦɟɧɬɧɵɯ
ɩɨɞɦɧɨɠɟɫɬɜ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ).
v=prompt('ȼɜɟɞɢ n ɢ k<n','6,4').split(','); k=+v[1]; n=+v[0];
a=k; x=0; g=''; j=0; for(i=0; i<=n; i++)v[i]=i;
while(a>0){g+=v[1]; for(i=2; i<=k; i++)g+=','+v[i]; g+='\n'; j++
if(k==n)break; if(v[k]==n)a--; else a=k;
if(a>0){x=v[a]; for(i=a; i<=k; i++)v[i]=x-a+i+1; }}j+'\n'+g
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 10 ɞɥɹ n = 4 ɢ k = 4
ȼɫɟɝɨ ɩɨɞɦɧɨɠɟɫɬɜ 15:
<1,2,3,4>
<1,2,3,5>
<1,2,3,6>
<1,2,4,5>
<1,2,4,6>
<1,2,5,6>
<1,3,4,5>
<1,3,4,6>
<1,3,5,6>
<1,4,5,6>
<2,3,4,5>
<2,3,4,6>
<2,3,5,6>
<2,4,5,6>
<3,4,5,6>
24
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɂɚɦɟɬɢɦ, ɱɬɨ ɢɡ ɩɨɞɦɧɨɠɟɫɬɜ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ, ɡɚɩɢɫɚɧɧɵɯ
ɜ ɜɢɞɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɟɣ ɞɥɢɧɵ n ɢ ɫɨɫɬɨɹɳɢɯ ɢɡ ɧɭɥɟɣ ɢ ɟɞɢɧɢɰ, ɦɨɠɧɨ
L
ɩɨɫɬɪɨɢɬɶ n-ɦɟɪɧɨɟ ɥɢɧɟɣɧɨɟ ɧɨɪɦɢɪɨɜɚɧɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ
Z
= {0, 1}. Ɉɩɟɪɚɰɢɟɣ ɫɥɨɠɟɧɢɹ ɜɟɤɬɨɪɨɜ, ɧɚɩɪɢɦɟɪ, ɩɨɤɨɨɪɞɢɧɚɬɧɚɹ ɞɢɡɴ-
2
ɧɚɞ ɩɨɥɟɦ
n
ɸɧɤɰɢɹ, ɬɨ ɟɫɬɶ ɩɨ ɩɪɚɜɢɥɚɦ: 1 + 0 = 1, 1 + 1 = 1, 0 + 0 = 0. Ⱦɥɹ n = 5,
ɚ = (1, 0, 1, 0, 1) ɢ b = (1, 0, 1, 1, 1) ɫɭɦɦɚ a + b = (1, 0, 0, 1, 1) – ɨɛɴɟɞɢɧɟɧɢɟ
L
ɩɨɞɦɧɨɠɟɫɬɜ ɚ ɢ b. ɇɨɪɦɨɣ |ɚ| ɷɥɟɦɟɧɬɚ ɚ
ɹɜɥɹɟɬɫɹ ɦɨɳɧɨɫɬɶ ɷɬɨɝɨ ɩɨɞ-
n
ɦɧɨɠɟɫɬɜɚ ɢɥɢ ɩɪɨɢɡɜɟɞɟɧɢɟ ɚ·ɚ = |ɚ|, ɝɞɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɚ·b – ɨɛɵɱɧɨɟ ɫɤɚ-
ɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɨɜ ɚ ɢ b. ɇɟɫɥɨɠɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɚ·b ɪɚɜɧɨ ɦɨɳɧɨɫɬɢ ɩɟɪɟɫɟɱɟɧɢɹ ɩɨɞɦɧɨɠɟɫɬɜ ɚ ɢ b. Ɉɞɧɚɤɨ ɫɜɨɣɫɬɜɨ ɫɤɚɥɹɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ: ɚ·(b + c) = ɚ·b + ɚ·c, ɜɵɩɨɥɧɹɟɬɫɹ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ
b·c = 0. ɉɨɞɭɦɚɣɬɟ
ɤɚɤ ɩɨɫɬɪɨɢɬɶ ɬɚɤɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ.
Ɍɟɩɟɪɶ ɧɚɫɬɚɥɨ ɜɪɟɦɹ ɡɚɧɹɬɶɫɹ ɩɨɜɬɨɪɟɧɢɟɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɞɢɫɰɢɩɥɢ-
ɧɵ: «Ⱥɥɝɟɛɪɚ».
§ 2. Ƚɪɭɩɩɵ
ɇɚɱɧɟɦ ɫ ɨɫɧɨɜɧɵɯ ɩɨɧɹɬɢɣ. ɉɭɫɬɶ G – ɦɧɨɠɟɫɬɜɨ ɨɛɴɟɤɬɨɜ ɥɸɛɨɣ
ɩɪɢɪɨɞɵ. ɋɸɪɴɟɤɬɢɜɧɚɹ ɮɭɧɤɰɢɹ
o
G×G
G
ɧɚɡɵɜɚɟɬɫɹ ɢɧɨɝɞɚ ɡɚɤɨɧɨɦ ɤɨɦɩɨɡɢɰɢɢ (ɧɚ G ɜ ɫɟɛɹ). ȿɫɥɢ ɯ ɢ ɭ – ɷɥɟɦɟɧɬɵ ɢɡ
G, ɬɨ ɨɛɪɚɡ ɩɚɪɵ (ɯ, ɭ) ɩɪɢ ɷɬɨɦ ɨɬɨɛɪɚɠɟɧɢɢ ɱɚɫɬɨ ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɨɟ ɨɛɨɡɧɚ-
ɱɟɧɢɟ ɢ ɧɚɡɵɜɚɸɬ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɚɤɨɧɚ ɤɨɦɩɨɡɢɰɢɢ
ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ ɯɭ ɢɥɢ ɯǜɭ. Ⱥ ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɭɞɨɛɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɚɞɞɢɬɢɜɧɨɟ
ɨɛɨɡɧɚɱɟɧɢɟ ɢ ɩɢɫɚɬɶ ɯ
+ ɭ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɷɥɟɦɟɧɬ ɯ + ɭ ɧɚɡɵɜɚɸɬ ɫɭɦɦɨɣ ɯ ɢ ɭ.
Ɉɛɵɱɧɨ ɨɛɨɡɧɚɱɟɧɢɟ ɯ + ɭ ɢɫɩɨɥɶɡɭɸɬ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɜɵɩɨɥɧɹɟɬɫɹ
ɫɨɨɬɧɨɲɟɧɢɟ ɯ + ɭ = ɭ + ɯ. Ɍɚɤɢɟ ɝɪɭɩɩɵ ɧɚɡɵɜɚɸ ɚɛɟɥɟɜɵɦɢ.
ɉɭɫɬɶ G – ɦɧɨɠɟɫɬɜɨ, ɧɚɞɟɥɟɧɧɨɟ ɡɚɤɨɧɨɦ ɤɨɦɩɨɡɢɰɢɢ. ɉɪɨɢɡɜɟɞɟɧɢɟ
ɷɥɟɦɟɧɬɨɜ ɯ, ɭ, z ɢɡ G ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ
ɞɜɭɦɹ ɫɩɨɫɨɛɚɦɢ: (ɯɭ)z ɢ ɯ(ɭz). ȿɫɥɢ
(ɯɭ)z = ɯ(ɭz) ɞɥɹ ɜɫɟɯ ɯ, ɭ, z ɢɡ G, ɬɨ ɝɨɜɨɪɹɬ, ɱɬɨ ɡɚɤɨɧ ɤɨɦɩɨɡɢɰɢɢ ɚɫɫɨɰɢɚɬɢɜɟɧ. ȿɫɥɢ ɞɥɹ ɜɫɟɯ ɯ, ɭ ɢɡ G ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ: ɯɭ = ɭɯ, ɬɨ ɡɚɤɨɧ ɤɨɦ-
ɩɨɡɢɰɢɢ ɤɨɦɦɭɬɚɬɢɜɟɧ.
ɗɥɟɦɟɧɬ ɟ ɢɡ G, ɬɚɤɨɣ, ɱɬɨ ɯɟ
= ɟɯ = ɯ ɞɥɹ ɜɫɟɯ ɯG, ɧɚɡɵɜɚɟɬɫɹ ɟɞɢɧɢɱ-
ɧɵɦ ɷɥɟɦɟɧɬɨɦ (ɢɥɢ ɟɞɢɧɢɰɟɣ, ɢ ɨɛɨɡɧɚɱɚɸɬ 1). Ʉɨɝɞɚ ɡɚɤɨɧ ɤɨɦɩɨɡɢɰɢɢ ɡɚ-
ɩɢɫɵɜɚɟɬɫɹ ɚɞɞɢɬɢɜɧɨ, ɟɞɢɧɢɱɧɵɣ ɷɥɟɦɟɧɬ ɨɛɨɡɧɚɱɚɟɬɫɹ ɱɟɪɟɡ 0 ɢ ɧɚɡɵɜɚɟɬɫɹ
ɧɭɥɟɜɵɦ ɷɥɟɦɟɧɬɨɦ.
ȿɞɢɧɢɱɧɵɣ ɷɥɟɦɟɧɬ ɟɞɢɧɫɬɜɟɧ, ɩɨɫɤɨɥɶɤɭ, ɟɫɥɢ ɟ' – ɞɪɭɝɨɣ ɟɞɢɧɢɱɧɵɣ
ɷɥɟɦɟɧɬ, ɬɨ, ɩɨ ɩɪɟɞɩɨɥɨɠɟɧɢɸ, ɢɦɟɟɦ
ɟ = ɟɟ' = ɟ'.
Ɇɨɧɨɢɞ – ɷɬɨ ɦɧɨɠɟɫɬɜɨ G ɫ ɚɫɫɨɰɢɚɬɢɜɧɵɦ ɡɚɤɨɧɨɦ ɤɨɦɩɨɡɢɰɢɢ, ɨɛ-
ɥɚɞɚɸɳɢɦ ɟɞɢɧɢɱɧɵɦ ɷɥɟɦɟɧɬɨɦ (ɬɚɤ ɱɬɨ, ɜ ɱɚɫɬɧɨɫɬɢ, G ɧɟ ɩɭɫɬɨ).
25
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ƚɪɭɩɩɚ G – ɷɬɨ ɦɨɧɨɢɞ, ɜ ɤɨɬɨɪɨɦ ɞɥɹ ɤɚɠɞɨɝɨ ɷɥɟɦɟɧɬɚ ɯG ɫɭɳɟɫɬɜɭ-
ɟɬ ɷɥɟɦɟɧɬ ɭG, ɬɚɤɨɣ, ɱɬɨ ɯɭ = ɭɯ = ɟ. ɗɥɟɦɟɧɬ ɭ ɧɚɡɵɜɚɟɬɫɹ ɨɛɪɚɬɧɵɦ ɤ ɯ.
Ɉɛɪɚɬɧɵɣ ɷɥɟɦɟɧɬ ɟɞɢɧɫɬɜɟɧ; ɞɟɣɫɬɜɢɬɟɥɶɧɨ, ɟɫɥɢ ɭ' – ɞɪɭɝɨɣ ɨɛɪɚɬɧɵɣ ɤ ɯ,
ɬɨ
ɭ' = ɭ'ɟ = ɭ'(ɯɭ) = (ɭ'ɯ)ɭ = ɟɭ = ɭ.
Ɉɛɨɡɧɚɱɚɟɬɫɹ ɷɬɨɬ
ɷɥɟɦɟɧɬ ɱɟɪɟɡ ɯ-1 (ɢɥɢ –ɯ, ɤɨɝɞɚ ɡɚɤɨɧ ɤɨɦɩɨɡɢɰɢɢ
ɡɚɩɢɫɵɜɚɟɬɫɹ ɚɞɞɢɬɢɜɧɨ).
Ɇɵ ɦɨɝɥɢ ɛɵ ɬɚɤɠɟ ɨɩɪɟɞɟɥɢɬɶ ɥɟɜɵɟ ɟɞɢɧɢɰɵ ɢ ɥɟɜɵɟ ɨɛɪɚɬɧɵɟ
ɷɥɟɦɟɧɬɵ (ɨɱɟɜɢɞɧɵɦ ɫɩɨɫɨɛɨɦ). ɇɨ ɥɟɝɤɨ ɞɨɤɚɡɚɬɶ, ɱɬɨ ɨɧɢ ɹɜɥɹɸɬɫɹ ɧɚ ɫɚɦɨɦ ɞɟɥɟ ɟɞɢɧɢɰɚɦɢ ɢ ɨɛɪɚɬɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ. ɂɦɟɧɧɨ ɜɟɪɧɨ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ:
ɍɬɜɟɪɠɞɟɧɢɟ 1. ɉɭɫɬɶ – ɦɧɨɠɟɫɬɜɨ G – ɦɧɨɠɟɫɬɜɨ ɫ ɚɫɫɨɰɢɚɬɢɜɧɵɦ
ɡɚɤɨɧɨɦ ɤɨɦɩɨɡɢɰɢɢ, ɟ – ɥɟɜɚɹ
ɟɞɢɧɢɰɚ ɷɬɨɝɨ ɦɧɨɠɟɫɬɜɚ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ
ɭ ɤɚɠɞɨɝɨ ɷɥɟɦɟɧɬɚ G ɟɫɬɶ ɥɟɜɵɣ ɨɛɪɚɬɧɵɣ. Ɍɨɝɞɚ ɟ – ɟɞɢɧɢɰɚ ɢ ɜɫɹɤɢɣ ɥɟɜɵɣ ɨɛɪɚɬɧɵɣ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɨɛɪɚɬɧɵɦ.
Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɪɚɫɫɦɨɬɪɢɦ ɩɪɨɢɡɜɨɥɶɧɵɣ ɷɥɟɦɟɧɬ ɯG ɢ ɭG – ɟɝɨ
ɥɟɜɵɣ ɨɛɪɚɬɧɵɣ. Ɍɨɝɞɚ ɭɯ = ɟ, ɨɬɫɸɞɚ
ɭɯɭ = ɟɭ = ɭ.
ɍɦɧɨɠɚɹ ɷɬɨ ɪɚɜɟɧɫɬɜɨ (ɭ = ɭɯɭ) ɫɥɟɜɚ ɧɚ
zG – ɥɟɜɵɣ ɨɛɪɚɬɧɵɣ ɞɥɹ ɭ,
ɩɨɥɭɱɢɦ
ɟ = zɭ = z(ɭɯɭ) = (zɭ)(ɯɭ) = ɟ(ɯɭ) = ɯɭ,
ɬɨ ɟɫɬɶ ɭ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɩɪɚɜɵɦ ɨɛɪɚɬɧɵɦ ɤ ɯ. Ʉɪɨɦɟ ɬɨɝɨ,
ɯɟ = ɯ(ɭɯ) = (ɯɭ)ɯ = ɟɯ = ɯ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɟ – ɩɪɚɜɚɹ ɟɞɢɧɢɰɚ. ɍɬɜɟɪɠɞɟɧɢɟ ɞɨɤɚɡɚɧɨ.
ɉɭɫɬɶ G – ɝɪɭɩɩɚ. ɉɨɞɝɪɭɩɩɨɣ ɇ ɝɪɭɩɩɵ G ɧɚɡɵɜɚɟɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɨ
G, ɫɨɞɟɪɠɚɳɟɟ ɟɞɢɧɢɱɧɵɣ ɷɥɟɦɟɧɬ, ɢ ɡɚɦɤɧɭɬɨɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɚɤɨɧɚ ɤɨɦ-
ɜ
ɩɨɡɢɰɢɢ (ɬɨ ɟɫɬɶ ɩɨɞɦɨɧɨɢɞ) ɢ ɜɡɹɬɢɹ ɨɛɪɚɬɧɨɝɨ ɷɥɟɦɟɧɬɚ (ɬɨ ɟɫɬɶ ɯ
–1
G, ɟɫ-
ɥɢ ɯG). ɉɨɞɝɪɭɩɩɚ ɧɚɡɵɜɚɟɬɫɹ ɬɪɢɜɢɚɥɶɧɨɣ, ɟɫɥɢ ɨɧɚ ɫɨɫɬɨɢɬ ɢɡ ɨɞɧɨɝɨ
ɟɞɢɧɢɱɧɨɝɨ ɷɥɟɦɟɧɬɚ.
Ɉɱɟɜɢɞɧɨ, ɩɟɪɟɫɟɱɟɧɢɟ ɥɸɛɨɝɨ ɫɟɦɟɣɫɬɜɚ ɩɨɞɝɪɭɩɩ ɟɫɬɶ ɩɨɞɝɪɭɩɩɚ (ɞɨ-
ɤɚɠɢɬɟ ɷɬɨ).
ɇG – ɩɨɞɝɪɭɩɩɚ ɝɪɭɩɩɵ G ɢ ɯG. Ɇɧɨɠɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɜɢɞɚ ɯɇ ɧɚ-
ɡɵɜɚɟɬɫɹ ɥɟɜɵɦ ɫɦɟɠɧɵɦ ɤɥɚɫɫɨɦ ɝɪɭɩɩɵ G ɩɨ ɇ. ȼɫɹɤɢɣ ɷɥɟɦɟɧɬ ɢɡ ɯɇ
ɧɚ-
ɡɵɜɚɟɬɫɹ ɩɪɟɞɫɬɚɜɢɬɟɥɟɦ ɫɦɟɠɧɨɝɨ ɤɥɚɫɫɚ ɯɇ.
Ɉɬɨɛɪɚɠɟɧɢɟ ɭ հɯɭ ɫɸɪɴɟɤɬɢɜɧɨ, ɩɨɷɬɨɦɭ ɥɸɛɵɟ ɞɜɚ ɥɟɜɵɯ ɫɦɟɠɧɵɯ
ɤɥɚɫɫɚ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɭɸ ɦɨɳɧɨɫɬɶ. Ʉɪɨɦɟ ɬɨɝɨ, ɟɫɥɢ ɞɜɚ ɫɦɟɠɧɵɯ ɤɥɚɫɫɚ
26
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɯɇ ɢ ɭɇ, ɢɦɟɸɳɢɟ ɯɨɬɹ ɛɵ ɨɞɢɧ ɨɛɳɢɣ ɷɥɟɦɟɧɬ, ɫɨɜɩɚɞɚɸɬ. ȼ ɫɚɦɨɦ ɞɟɥɟ,
ɩɭɫɬɶ ɫɭɳɟɫɬɜɭɟɬ zɯɇ ɢ zɭɇ, ɬɨɝɞɚ z = ɯh = yh
ɜɚ ɧɚ ɨɛɪɚɬɧɵɣ ɤ h ɷɥɟɦɟɧɬ h
ɬɟɥɶɧɨ, ɯɇ = y(h
–1
h
)ɇ = yɇ, ɩɨɬɨɦɭ ɱɬɨ ɥɸɛɨɝɨ hɇ ɢɦɟɟɦ hɇ = ɇ.
1
–1
ɇ, ɩɨɥɭɱɢɦ ɯ = y(h1h
, ɝɞɟ h, h1ɇ. ɍɦɧɨɠɚɹ ɫɩɪɚ-
1
–1
), ɧɨ h1h
–1
ɇ, ɫɥɟɞɨɜɚ-
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, G ɟɫɬɶ ɨɛɴɟɞɢɧɟɧɢɟ ɩɨɩɚɪɧɨ ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɫɦɟɠɧɵɯ ɤɥɚɫɫɨɜ ɩɨ ɇ. Ⱥɧɚɥɨɝɢɱɧɨɟ ɡɚɦɟɱɚɧɢɟ ɩɪɢɦɟɧɢɦɨ ɤ ɩɪɚɜɵɦ ɫɦɟɠɧɵɦ
ɤɥɚɫɫɚɦ (ɬɨ ɟɫɬɶ ɩɨɞɦɧɨɠɟɫɬɜɚɦ ɜ G ɜɢɞɚ ɇɯ). ɑɢɫɥɨ ɥɟɜɵɯ ɫɦɟɠɧɵɯ ɤɥɚɫɫɨɜ
ɝɪɭɩɩɵ G ɩɨ ɇ ɨɛɨɡɧɚɱɚɟɬɫɹ ɱɟɪɟɡ (G:ɇ) ɢ ɧɚɡɵɜɚɟɬɫɹ
(ɥɟɜɵɦ) ɢɧɞɟɤɫɨɦ ɩɨɞɝɪɭɩɩɵ ɇ ɜ G. ɂɧɞɟɤɫ ɬɪɢɜɢɚɥɶɧɨɣ ɩɨɞɝɪɭɩɩɵ ɧɚɡɵɜɚɟɬɫɹ ɩɨɪɹɞɤɨɦ ɝɪɭɩɩɵ
G ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ (G:1).
Ɍɟɨɪɟɦɚ 1 (ɬɟɨɪɟɦɚ Ʌɚɝɪɚɧɠɚ). ɉɭɫɬɶ G – ɝɪɭɩɩɚ ɢ ɇ – ɟɟ ɩɨɞɝɪɭɩɩɚ.
Ɍɨɝɞɚ
(G:ɇ) (ɇ:1) = (G:1).
ȼ ɬɨɦ ɫɦɵɫɥɟ, ɱɬɨ ɟɫɥɢ ɞɜɚ ɢɡ ɷɬɢɯ ɢɧɞɟɤɫɨɜ ɤɨɧɟɱɧɵ, ɬɨ ɤɨɧɟɱɟɧ
ɦɟɫɬɨ ɧɚɩɢɫɚɧɧɨɟ ɪɚɜɟɧɫɬɜɨ. ȿɫɥɢ ɩɨɪɹɞɨɤ ɝɪɭɩɩɵ (G:1)
ɢ ɬɪɟɬɢɣ, ɢ ɢɦɟɟɬ
ɤɨɧɟɱɟɧ, ɬɨ ɨɧ ɞɟɥɢɬɫɹ ɧɚ ɩɨɪɹɞɨɤ ɩɨɞɝɪɭɩɩɵ ɇ.
Ȼɨɥɟɟ ɨɛɳɨ, ɟɫɥɢ K ɇ G – ɝɪɭɩɩɵ, ɬɨ
(G:K) = (G:ɇ) (ɇ:K).
ȿɫɥɢ ɩɨɞɝɪɭɩɩɚ ɇ ɝɪɭɩɩɵ G ɬɚɤɨɜɚ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ ɯG ɜɵɩɨɥɧɹɟɬɫɹ
ɪɚɜɟɧɫɬɜɨ: ɯɇ = ɇɯ, ɬɨ ɬɚɤɚɹ ɩɨɞɝɪɭɩɩɚ ɧɚɡɵɜɚɟɬɫɹ
ɧɨɪɦɚɥɶɧɨɣ
.
Ʌɟɝɤɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ ɩɨɞɝɪɭɩɩɚ ɇ ɝɪɭɩɩɵ G ɧɨɪɦɚɥɶɧɚ, ɬɨ ɦɧɨɠɟɫɬɜɨ ɟɟ ɫɦɟɠɧɵɯ ɤɥɚɫɫɨɜ ɨɛɪɚɡɭɸɬ ɝɪɭɩɩɭ ɫ ɧɟɣɬɪɚɥɶɧɵɦ ɷɥɟɦɟɧɬɨɦ ɇ.
ȼ ɫɚɦɨɦ ɞɟɥɟ, ɟɫɥɢ ɯɇ ɢ ɭɇ – ɫɦɟɠɧɵɟ ɤɥɚɫɫɵ, ɬɨ ɯɇɭɇ = ɯɭɇɇ = ɯɭɇ, ɬɨ ɟɫɬɶ
ɢɯ ɩɪɨɢɡɜɟɞɟɧɢɟ ɹɜɥɹɟɬɫɹ ɫɦɟɠɧɵɦ ɤɥɚɫɫɨɦ ɢ ɯ
–1
ɇ – ɨɛɪɚɬɧɵɣ ɤ ɯ–1ɇ.
Ƚɪɭɩɩɚ ɫɦɟɠɧɵɯ ɤɥɚɫɫɨɜ ɩɨ ɧɨɪɦɚɥɶɧɨɣ ɩɨɞɝɪɭɩɩɟ ɇ ɧɚɡɵɜɚɟɬɫɹ ɮɚɤɬɨɪɝɪɭɩɩɨɣ ɝɪɭɩɩɵ G ɩɨ ɇ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ G/ɇ, ɟɟ ɷɥɟɦɟɧɬɵ – ɤɚɤ ɷɥɟɦɟɧɬɵ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɫɦɟɠɧɵɯ ɤɥɚɫɫɨɜ.
ɉɭɫɬɶ ɯG. Ɍɨɝɞɚ, ɨɱɟɜɢɞɧɨ, ɱɬɨ ɯɯ = ɯ
ɟɫɬɶ ɩɨɞɦɧɨɠɟɫɬɜɨ {e, ɯ, ɯ
ɰɢɤɥɢɱɟɫɤɨɣ, ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɣ ɷɥɟɦɟɧɬ ɯG, ɱɬɨ ɜɫɹɤɢɣ ɷɥɟɦɟɧɬ ɭG
ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧ ɜ ɜɢɞɟ ɭ = ɯ
m
ɝɨ ɯ
= ɟ ɢ m > 0, ɬɨ m ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɩɨɤɚɡɚɬɟɥɟɦ ɷɥɟɦɟɧɬɚ ɯ.
–1
, ɯ2, ɯ–2, … } – ɩɨɞɝɪɭɩɩɚ G. Ƚɪɭɩɩɚ G ɧɚɡɵɜɚɟɬɫɹ
m
, ɝɞɟ mZ. ȿɫɥɢ m – ɰɟɥɨɟ ɱɢɫɥɨ ɞɥɹ ɤɨɬɨɪɨ-
ȿɫɥɢ ɝɪɭɩɩɚ ɤɨɧɟɱɧɚ, ɬɨ ɟɫɬɶ (G:1) = |G| = n ɢɥɢ G = {ɟ, a
ɭ ɤɚɠɞɨɝɨ ɟɟ ɷɥɟɦɟɧɬɚ ɯG ɫɭɳɟɫɬɜɭɟɬ ɩɨɤɚɡɚɬɟɥɶ k n (ɯ
ɩɨɞɦɧɨɠɟɫɬɜɨ ɇ = {e, x,…, x
k
} ɹɜɥɹɟɬɫɹ ɰɢɤɥɢɱɟɫɤɨɣ ɩɨɞɝɪɭɩɩɨɣ G. Ɉɬɫɸɞɚ
2
, ɯ–1ɯ
–1
= ɯ–2, ɯ3, ɯ–3, … G. Ɍɨ
, …, a
1
k
= e). ɉɪɢ ɷɬɨɦ
n–1
}, ɬɨ
ɢ ɬɟɨɪɟɦɵ Ʌɚɝɪɚɧɠɚ ɢɦɟɟɦ ɫɥɟɞɭɸɳɢɟ ɫɥɟɞɫɬɜɢɹ:
ɋɥɟɞɫɬɜɢɟ 1. ȿɫɥɢ G – ɝɪɭɩɩɚ ɤɨɧɟɱɧɨɝɨ ɩɨɪɹɞɤɚ, ɬɨ ɞɥɹ ɥɸɛɨɝɨ ɯG
n
|G|
= ɯ
ɯ
= e.
27
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɋɥɟɞɫɬɜɢɟ 2. Ʌɸɛɚɹ ɝɪɭɩɩɚ ɩɪɨɫɬɨɝɨ ɩɨɪɹɞɤɚ ɹɜɥɹɟɬɫɹ ɰɢɤɥɢɱɟɫɤɨɣ.
Ⱦɥɹ ɩɪɢɦɟɪɚ ɪɚɫɫɦɨɬɪɢɦ ɝɪɭɩɩɭ G – ɜɪɚɳɟɧɢɣ ɩɪɚɜɢɥɶɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ. ɉɨɪɹɞɨɤ ɷɬɨɣ ɝɪɭɩɩɵ ɪɚɜɟɧ 6. ɗɥɟɦɟɧɬɨɦ ɚ ɹɜɥɹɟɬɫɹ, ɧɚɩɪɢɦɟɪ, ɩɨɜɨɪɨɬ ɧɚ ɭɝɨɥ 120
ɫɤɭɸ ɩɨɞɝɪɭɩɩɭ ɇ = {a, a
ɜɨɡɶɦɟɦ ɩɨɜɨɪɨɬ ɧɚ 180
ɟɬ ɰɢɤɥɢɱɟɫɤɭɸ ɝɪɭɩɩɭ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ: {b, b
ɨ
ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ. ɗɬɨɬ ɷɥɟɦɟɧɬ ɚ ɨɛɪɚɡɭɟɬ ɰɢɤɥɢɱɟ-
2
, a3 = e} ɩɨɪɹɞɤɚ 3. ȼ ɤɚɱɟɫɬɜɟ ɞɪɭɝɨɝɨ ɷɥɟɦɟɧɬɚ b
ɨ
ɜɨɤɪɭɝ ɜɵɫɨɬɵ ɬɪɟɭɝɨɥɶɧɢɤɚ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɨɛɪɚɡɭ-
2
= e}:
Ʌɟɜɵɣ ɫɦɟɠɧɵɣ ɤɥɚɫɫ ɩɨ ɩɨɞɝɪɭɩɩɟ H: bH = {b, ba, ba
ɫɦɟɠɧɨɦɭ ɤɥɚɫɫɭ Hb = {b, ab, a
2
b}, ɯɨɬɹ ab z ba (ab = ba2). ɗɥɟɦɟɧɬɵ a ɢ b
ɧɚɡɵɜɚɸɬ ɨɛɪɚɡɭɸɳɢɦɢ ɝɪɭɩɩɵ G, ɚ G = {e, a, a
2
, b, ab, ba}.
2
}, ɪɚɜɟɧ ɩɪɚɜɨɦɭ
ɇɚɣɞɢɬɟ ɝɪɭɩɩɭ ɜɪɚɳɟɧɢɣ ɩɪɚɜɢɥɶɧɨɝɨ ɬɟɬɪɚɷɞɪɚ. ɍɛɟɞɢɬɟɫɶ, ɱɬɨ ɜ ɧɟɣ
12 ɷɥɟɦɟɧɬɨɜ, ɧɚɣɞɢɬɟ ɟɟ ɩɨɞɝɪɭɩɩɵ ɢ ɨɛɪɚɡɭɸɳɢɟ.
Ɇɧɨɠɟɫɬɜɨ ɰɟɥɵɯ ɱɢɫɟɥ Z = {0, ±1, ±2, … } ɹɜɥɹɟɬɫɹ ɰɢɤɥɢɱɟɫɤɨɣ ɚɞɞɢɬɢɜɧɨɣ ɝɪɭɩɩɨɣ (ɫ ɨɩɟɪɚɰɢɟɣ – ɨɛɵɱɧɨɟ ɫɥɨɠɟɧɢɟ). Ⱦɨɤɚɠɢɬɟ ɷɬɨ. ȿɟ ɩɨɞɝɪɭɩɩɚɦɢ ɹɜɥɹɸɬɫɹ, ɧɚɩɪɢɦɟɪ: mZ, m = 0, 1, … (ɩɪɢ m = 0 – ɬɪɢɜɢɚɥɶɧɚɹ ɩɨɞɝɪɭɩɩɚ, m = 1 – ɫɚɦɚ ɝɪɭɩɩɚ Z, m
= 2 – ɜɫɟ ɱɟɬɧɵɟ ɰɟɥɵɟ ɱɢɫɥɚ, m = 3 – ɝɪɭɩɩɚ
ɰɟɥɵɯ ɱɢɫɟɥ ɤɪɚɬɧɵɯ 3 ɢ ɬ.ɞ. ɉɨɱɟɦɭ ɦɵ ɧɟ ɭɤɚɡɚɥɢ ɜ ɷɬɨɦ ɫɩɢɫɤɟ, ɧɚɩɪɢɦɟɪ,
ɩɨɞɝɪɭɩɩɭ (–3Z)? Ⱦɚ ɩɨɬɨɦɭ ɱɬɨ (–3Z) = 3Z). Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɜɫɟ ɷɬɢ ɩɨɞɝɪɭɩɩɵ ɧɨɪɦɚɥɶɧɵɟ ɢ ɰɢɤɥɢɱɟɫɤɢɟ (ɞɨɤɚɠɢɬɟ ɷɬɨ).
Ⱦɨɤɚɠɟɦ, ɱɬɨ ɞɪɭɝɢɯ ɩɨɞɝɪɭɩɩ ɜ Z ɧɟ ɫɭɳɟɫɬɜɭɟɬ. ȼ ɫɚɦɨɦ ɞɟɥɟ, ɩɭɫɬɶ
ɇZ – ɩɨɞɝɪɭɩɩɚ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɇ ɫɭɳɟɫɬɜɭɟɬ ɦɢɧɢɦɚɥɶɧɵɣ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɷɥɟɦɟɧɬ (1 ɟɫɬɶ ɦɢɧɢɦɚɥɶɧɵɣ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɷɥɟɦɟɧɬ ɜ Z). Ɉɛɨɡɧɚɱɢɦ
ɷɬɨɬ ɷɥɟɦɟɧɬ ɱɟɪɟɡ ɚ. Ɍɨɝɞɚ, ɨɱɟɜɢɞɧɨ, ɱɬɨ ɚZ = {0, ±ɚ, ±2ɚ, … } Z ɩɨɞɝɪɭɩɩɚ.
ȼɨɡɶɦɟɦ ɩɪɨɢɡɜɨɥɶɧɵɣ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɷɥɟɦɟɧɬ ɯɇ ɢ ɪɚɡɞɟɥɢɦ ɟɝɨ ɧɚ ɚ, ɩɨɥɭɱɢɦ
ɯ = ɪɚ + r,
ɝɞɟ r – ɨɫɬɚɬɨɤ ɨɬ ɞɟɥɟɧɢɹ ɯ ɧɚ ɚ (0 r < ɚ, ɪZ+).
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ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɉɬɫɸɞɚ
r = ɯ – ɪɚ = ɯ + ɪ(–ɚ),
ɬɨ ɟɫɬɶ r ɹɜɥɹɟɬɫɹ ɷɥɟɦɟɧɬɨɦ ɝɪɭɩɩɵ ɇ, ɧɨ ɬɨɝɞɚ r = 0 (ɬɚɤ ɤɚɤ ɚ – ɦɢɧɢɦɚɥɶɧɵɣ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɜ ɇ) ɢ ɇ = ɚZ.
Ɏɚɤɬɨɪ ɝɪɭɩɩɵ Z ɩɨ mZ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɤɚɤ Z
= {0, 1, …, m – 1},
m·Zm
ɬɚɤ ɤɚɤ 0, 1, …, m – 1 – ɩɪɟɞɫɬɚɜɢɬɟɥɢ ɫɦɟɠɧɵɯ ɤɥɚɫɫɨɜ mZ, 1 + mZ, …,
m – 1 + mZ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
Ɋɚɫɫɦɨɬɪɢɦ Z
ɛɨɥɟɟ ɩɨɞɪɨɛɧɨ. ɋɧɚɱɚɥɚ ɡɚɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ
m
d = ɇɈȾ(m, n) – ɧɚɢɛɨɥɶɲɢɣ ɨɛɳɢɣ ɞɟɥɢɬɟɥɶ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ m ɢ n, ɬɨ
mZ + nZ = dZ.
Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɧɚɞɨ ɞɨɤɚɡɚɬɶ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɪɟɲɟ-
ɧɢɹ ɞɢɨɮɚɧɬɨɜɚ ɭɪɚɜɧɟɧɢɹ:
ɏm + yn = d, (5)
ɬɨ ɟɫɬɶ ɭɪɚɜɧɟɧɢɹ, ɪɟɲɟɧɢɟɦ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɰɟɥɵɟ ɱɢɫɥɚ. ɗɬɨɬ ɮɚɤɬ ɫɥɟɞɭɟɬ ɫɪɚɡɭ ɠɟ
ɢɡ ɚɥɝɨɪɢɬɦɚ ȿɜɤɥɢɞɚ.
Ⱥɥɝɨɪɢɬɦ 11 (ɚɥɝɨɪɢɬɦ ȿɜɤɥɢɞɚ).
v=prompt('ȼɟɫɬɢ m ɢ n','22,33').split(','); m=+v[0]; n=+v[1];
t='ɇɈȾ('+m+','+n+')='; while(n){x=n; n=m%n; m=x}t+=x
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 11 ɞɥɹ n = 22 ɢ m = 33
ɇɈȾ(22, 33) = 11
Ɉɞɧɚɤɨ ɷɬɢ ɪɚɫɫɭɠɞɟɧɢɹ ɧɟ ɪɟɲɚɸɬ ɧɚɲɟɝɨ ɞɢɨɮɚɧɬɨɜɚ ɭɪɚɜɧɟɧɢɹ (5).
Ⱥɥɝɨɪɢɬɦ, ɪɟɲɚɸɳɢɣ ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɧɚɡɵɜɚɟɬɫɹ ɨɛɨɛɳɟɧɧɵɦ (ɭɥɭɱɲɟɧɧɵɦ,
ɪɚɫɲɢɪɟɧɧɵɦ) ɚɥɝɨɪɢɬɦɨɦ ȿɜɤɥɢɞɚ. ɉɪɢɜɟɞɟɦ ɨɞɢɧ ɢɡ ɬɚɤɢɯ ɚɥɝɨɪɢɬɦɨɜ:
Ⱥɥɝɨɪɢɬɦ 12 (ɨɛɨɛɳɟɧɧɵɣ ɚɥɝɨɪɢɬɦ ȿɜɤɥɢɞɚ).
u=prompt('ȼɜɟɞɢ ɚ ɢ b','14,93').split(',');
if(u[0]-u[1]<0){z=u[0]; u[0]=u[1]; u[1]=z}a=+u[1]; b=+u[0];
if(b==0){t='d='+a+' x=1 y=0'}else{x2=1; x1=0; y2=0; y1=1;
while(b){q=floor(a/b); r=a–q*b; x=x2–q*x1; y=y2–q*y1; a=b; b=r; x2=x1; x1=x;
y2=y1; y1=y}}
t='('+x2+')*'+u[1]+' + ('+y2+')*'+u[0]+' = '
t+=(x2*u[1]+y2*u[0])+' = ɇɈȾ('+u[1]+','+u[0]+')'
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 12 ɞɥɹ a = 14 ɢ b = 93
(20)*14 + (–3)*93 = 1 = ɇɈȾ(14, 93)
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ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɗɬɢ ɪɚɫɫɭɠɞɟɧɢɹ ɩɪɢɜɨɞɹɬ ɧɚɫ ɤ ɫɥɟɞɭɸɳɟɦɭ ɮɚɤɬɭ. ȿɫɥɢ ɪ – ɩɪɨ-
ɫɬɨɟ ɱɢɫɥɨ, ɬɨ ɩɨɞɦɧɨɠɟɫɬɜɨ {1, 2, …, p – 1} Z
ɹɜɥɹɟɬɫɹ ɝɪɭɩɩɨɣ ɩɨ ɨɩɟ-
ɪ
ɪɚɰɢɢ ɨɛɵɱɧɨɟ ɭɦɧɨɠɟɧɢɟ ɩɨ ɦɨɞɭɥɸ ɪ. ȼ ɫɚɦɨɦ ɞɟɥɟ, ɞɥɹ ɜɫɟɯ
ɯ ɢ y{1, 2, …, p – 1}
(ɯ + ɪZ)(ɭ + ɪZ) = ɯɭ + (ɭZ + ɯZ + ɪZ)ɪ = ɯɭ + ɪZ 0 + ɪZ,
ɬɚɤ ɤɚɤ ɇɈȾ(ɯ, ɪ) = ɇɈȾ(ɭ, ɪ) = 1 ɢ ɪ
– ɩɪɨɫɬɨɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, {1, 2, …,
p – 1} – ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɵɣ ɦɨɧɨɢɞ. Ɉɛɪɚɬɧɵɦ ɤ ɯ{1, 2, …, p – 1}, ɛɭɞɟɬ,
ɨɱɟɜɢɞɧɨ, ɬɚɤɨɣ ɭ{1, 2, …, p – 1}, ɱɬɨ (ɯy)mod p = 1, ɚ ɜɫɟ ɨɫɬɚɬɤɢ ɨɬ ɞɟɥɟ-
ɧɢɹ ɧɚ ɪ ɩɪɟɞɫɬɚɜɥɟɧɵ ɦɧɨɠɟɫɬɜɨɦ {0, 1, 2, …, p – 1}, ɩɨɷɬɨɦɭ ɨɧ ɫɭɳɟɫɬɜɭɟɬ
ɢ ɧɚɣɬɢ ɷɥɟɦɟɧɬ ɭ ɦɨɠɧɨ, ɧɚɩɪɢɦɟɪ, ɬɚɤ. ȼɵɩɢɲɟɦ p–1 ɨɫɬɚɬɤɨɜ ɨɬ ɞɟɥɟɧɢɹ
ɧɚ p ɫɥɟɞɭɸɳɢɯ ɱɢɫɟɥ: ɯ, ɯ
ɬɚɤ ɤɚɤ ɪ – ɩɪɨɫɬɨɟ ɱɢɫɥɨ. ȿɫɥɢ ɫɪɟɞɢ ɧɢɯ ɜɫɬɪɟɬɢɬɫɹ 1 = ɯ
2
, …, ɯ
p–1
. ȼɫɟ ɨɧɢ ɢɡ ɦɧɨɠɟɫɬɜɚ {1, 2, …, p – 1},
j
mod p, ɬɨ ɭ = ɯ
j–1
mod p, ɟɫɥɢ ɠɟ ɧɟɬ, ɬɨ, ɜ ɫɢɥɭ ɩɪɢɧɰɢɩɚ Ⱦɢɪɢɯɥɟ (ɩɪɢɧɰɢɩ Ⱦɢɪɢɯɥɟ: ɟɫɥɢ
(n + 1) ɤɪɨɥɢɤɨɜ ɩɨɦɟɫɬɢɥɢ ɜ n ɤɥɟɬɨɤ, ɬɨ ɜ ɧɟɤɨɬɨɪɨɣ ɤɥɟɬɤɟ ɨɤɚɠɟɬɫɹ, ɩɨ
ɤɪɚɣɧɟɣ ɦɟɪɟ, ɞɜɚ ɤɪɨɥɢɤɚ), ɫɪɟɞɢ ɧɢɯ ɛɭɞɟɬ ɞɜɚ ɨɞɢɧɚɤɨɜɵɯ, ɧɚɩɪɢɦɟɪ,
j
mod p ɢ ɯk mod p (k > j), ɧɨ ɬɨɝɞɚ ɯ
ɯ
ɪɚɬɧɵɣ ɤ ɯ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, {1, 2, …, p–1} Z
k–j
mod p = 1, ɬɨ ɟɫɬɶ ɭ = ɯ
– ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɚɹ ɤɨɦɦɭɬɚ-
ɪ
k–j–1
mod p – ɨɛ-
ɬɢɜɧɚɹ ɝɪɭɩɩɚ.
Ɍɟɨɪɟɦɚ 2 (ɦɚɥɚɹ ɬɟɨɪɟɦɚ Ɏɟɪɦɚ). ȿɫɥɢ ɰɟɥɨɟ ɱɢɫɥɨ ɚ ɧɟ ɞɟɥɢɬɫɹ ɧɚ
ɩɪɨɫɬɨɟ ɱɢɫɥɨ ɪ, ɬɨ ɱɢɫɥɨ (ɚ
ɪ–1 –
1) ɞɟɥɢɬɫɹ ɧɚ ɪ (ɚ
ɪ–1
mod n = 1).
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɷɬɨɣ ɬɟɨɪɟɦɵ ɩɪɨɜɟɞɢɬɟ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ.
Ɉɛɨɛɳɟɧɢɟɦ ɷɬɨɣ ɬɟɨɪɟɦɵ ɹɜɥɹɟɬɫɹ
Ɍɟɨɪɟɦɚ 3 (ɬɟɨɪɟɦɚ ɗɣɥɟɪɚ). ȿɫɥɢ m ɢ n ɜɡɚɢɦɧɨ ɩɪɨɫɬɵɟ ɱɢɫɥɚ. Ɍɨɝɞɚ
ɱɢɫɥɨ (m
ij(n)
– 1) ɞɟɥɢɬɫɹ ɧɚ n (m
ij(n)
mod n = 1), ɝɞɟ ij – ɮɭɧɤɰɢɹ ɗɣɥɟɪɚ:
ij(n) ؝ |{1 k < n : ɇɈȾ(k, n) = 1}|
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ɋɚɫɫɦɨɬɪɢɦ ɝɪɭɩɩɭ Z
ɝɞɟ b
, j = 2, …, ij(n), ɜɡɚɢɦɧɨ ɩɪɨɫɬɵɟ ɫ n. Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɬɟɨɪɟɦɵ ɞɨɫɬɚ-
j
ɬɨɱɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ {1, b
, …, b
2
ij(n)
(n > 1). ɉɭɫɬɶ {1, b2, …, b
n
ij(n)
}Zn,
} – ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɚɹ ɝɪɭɩɩɚ. Ⱥ ɞɥɹ ɷɬɨɝɨ
ɞɨɫɬɚɬɨɱɧɨ ɩɨɜɬɨɪɢɬɶ ɩɪɟɞɵɞɭɳɢɟ ɪɚɫɫɭɠɞɟɧɢɹ. Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
Ɉɬɦɟɬɢɦ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ ɮɭɧɤɰɢɢ ɗɣɥɟɪɚ.
ɋɜɨɣɫɬɜɚ ɮɭɧɤɰɢɢ ɗɣɥɟɪɚ.
1) ij(1) = 1;
2) ij(ɪ) = ɪ ɞɥɹ ɥɸɛɨɝɨ ɩɪɨɫɬɨɝɨ ɪ;
k
k–1
) = ɪ
3) ij(ɪ
(p – 1) ɞɥɹ ɥɸɛɨɝɨ ɩɪɨɫɬɨɝɨ ɪ ɢ ɧɚɬɭɪɚɥɶɧɨɝɨ k;
4) ij(ɪq) = ij(ɪ)ij(q) ɞɥɹ ɥɸɛɵɯ ɜɡɚɢɦɧɨ ɩɪɨɫɬɵɯ ɪ ɢ q.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɷɬɢɯ ɫɜɨɣɫɬɜ ɩɪɨɜɟɞɢɬɟ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ.
ȼ ɷɬɨɦ ɪɚɡɞɟɥɟ ɛɭɞɟɬ ɩɨɥɟɡɧɵɦ ɜɫɩɨɦɧɢɬɶ ɟɳɟ ɨɞɧɭ ɡɚɦɟɱɚɬɟɥɶɧɭɸ
ɬɟɨɪɟɦɭ, ɢɡɜɟɫɬɧɨɣ ɤɚɤ «ɤɢɬɚɣɫɤɚɹ ɬɟɨɪɟɦɚ ɨɛ ɨɫɬɚɬɤɚɯ».
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