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0.1.1. - ................................................................................... 18 0.1.2. .................................................................... 19 0.1.3. ................................................................... 20 0.1.4. ............................................................................... 20 0.1.5. ...................................................................................................... 21 0.1.6. ...................................................................... 21 0.1.7. ........................................................... 22

0.2. ................................................................................................................ 23 0.3. ................................................................................................... 24

0.3.1. ............................................................................. 24 0.3.2. ......................................................................................... 25 0.3.3. ................................................................................................. 25 0.3.4. .................................................................................................................... 26

0.4. = ++........................................................................................... 26 0.4.1. ................................................................................ 26 0.4.2. ................................................................... 27 0.4.3. C? .......................................................................................................................... 28 0.4.4. ............................................................................. 29

0.5. ........................................................................................................................... 29 ................................................................................................................................. 29 0.5.1. ................................................................................................................ 29 0.5.2. ........................................................................................................... 30 0.5.3. .................................................................................................... 30

1 ...................................................................... 33 1.1. ........................................................... 33

1.1.1. .............................................................................. 33 - ...........................................................................................................................33 - ....................................................................................................................................35 - . ...................................37 - ...........................................................................................................38 - , , n- ............................................................................................39 - . ........................................................................................40 - ...............................................................................................................................41

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1.1.5. , . ........................... 56 1.1.6. ................................................................................ 59

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1.1.7. .................................................................................................................. 65 - .............................................................65 - ....................................................................66

1.2. ............................................................................................................ 67 1.2.1. ....................................................................................................................... 68 1.2.2. ...................................................................................................... 69 1.2.3. - . ........................................................ 74 1.2.4. - ............................................................................................... 76 1.2.5. ....................................... 77

1.3. ................................................................................ 80 1.3.1. ..................................................................................................................... 80

- ...........................................................................................................................81 - .......................................................................................................84 - ...................................................................................................86

1.3.2. .......................................................................................................................... 86 - , .............................................................................................................86 - .............................................................................................................................88

1.3.3. ..................................................................................................................... 90 1.3.4. ......................................................................................................... 92

- ...........................92 - .........................93 - ...........................94

1.3.5. ............................................................................................. 96 - .....................................................................................................96

1.4. ................................................................................... 97 1.4.1. ............................................................................................ 99 1.4.2. ............................................................................................... 99 1.4.3. O(F): ..........................................................................................100 1.4.4. (F): .........................................................................................102 1.4.5. (F): .........................................................................................103 1.4.6. ..................................................................105 1.4.7. ..........................................................................106 1.4.8. ..................................................................107

- ....................................................................................................... 107 - ....................................................................................... 107 - ................................................................................................ 107 - if- ................................................................................................................... 107 - ................................................................................................................................. 108 - .................................................................................................................. 108 - ....................................................................................... 109 - ................................................................................................... 110 - ............................................................................................................................ 110

1.4.9. .....................................................................................112 - ............................................................. 112

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- .............................................................. 114 - ..................................................................................... 114

1.4.10. ........................................................117 - .................................................................................................. 117 - .................................................................................................... 118 - ............................................................................................................... 118

1.4.11. ..............................................................120 1.5. ...............................................................................................................120

1.5.1. ........................................................................................................120 1.5.2. ..................................................................................................................133

- , , .................................................................................... 133 - .................................................................................... 137 - ....................................................................................................... 138

2 ...............................................143 2.1. , , ........................................................................................................144

- .................................................................................................................................... 145 - ............................................................................................................................... 146 - ..................................................................................................................................... 147

2.1.1 () ..................................................................147 - .................................................................................................................................... 147 - ............................................................................................................................... 149

2.1.2 () .............................................................................151 - ....................................................................................................... 152 - ........................................................................................................ 153 - , ........................................................... 153 - , ........................................................... 154 - ....................................... 154 - , ................................................................ 155

2.2. ................................................................................................................158 - ..................................................................................................... 163 - ..................................................................................................... 164 - .................................................................................. 164 - ......................................................................................................................... 168

2.3. ........................................................................................................169 2.3.1. . - .............................................................................172 2.3.2. B- ......................................................................................................................174

2.4. - ........................................................................................................................176 - - ..................................................................................................................... 176 - ............................................................................................................................. 177

2.4.1. - .............................................................................................178 - ................................................................... 178 - .............................................................................................. 178 - - ................................................................................. 179 - - .......................................................... 179 - - .............................................................................. 180

2.4.2. .......................................................................................................184 - ............................................................................................................184

- ............................................................................................................. 184 - ...................................................................................................... 185 - .............................................................................................................. 185

- ..............................................................................................................185

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- ........................................................................................ 185 - .................................................................................................. 186

2.4.3. - .......................................................................................186 2.5. ...............................................................................................................192

2.5.1. ........................................................................................................192 2.5.2. ..................................................................................................................196

3 ...........................................................................................................199 3.1. ................................................................................................200

3.1.1. .................................................................................................200 3.1.2. ...........................................................................................201 3.1.3. . .............203 3.1.4. ....................................................................................................206 3.1.5. .............................................................................................207 3.1.6. ...........................................................................................208 3.1.7. .....................................................................213 3.1.8. ........................................................................................215 3.1.9. .........................................................................216 3.1.10. .....................219

3.2. ......................................................................................221 3.2.1. ......................................................................................221 3.2.2. ................................................................................................223 3.2.3. ..................................................................................................226 3.2.4. .....................................................................................229 3.2.5. ......................................................................................232

3.3. ........................................................................................................233 3.3.1. ...............................................................................235 3.3.2. .................................................................................236 3.3.3. ...............................................................................................237 3.3.4. ........................................................237 3.3.5. ..............................................................................................................237 3.3.6. .........................................................................................................238 3.3.7. .........................................................................238 3.3.8. - ...............................................................239 3.3.9. - .............................................................................239 3.3.10. ...............................................................................................239

3.4. ...............................................................................................................240 3.4.1. ........................................................................................................240 3.4.2. ..................................................................................................................243

4 .................................................................................................................245 4.1. .................................................................................................246

4.1.1. .........................................................248 4.1.2. ...........................................................249

4.2. . ..................................................................251 4.3. .................................................................................................................252 4.4. .......................................................................................................256 4.5. ..............................................................................................258 4.6. ...............................................................................................................260

4.6.1. ........................................................................................................260

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4.6.2. ..................................................................................................................261

5 .......................................................................................263 5.1. ...............................................................................................................263 5.2. ........................................................................267

5.2.1. .......................................................................................................267 5.2.2. , .......................................................268 5.2.3. ( ) .............................................268 5.2.4. ............................................269 5.2.5. ........................................................................................269 5.2.6. ...............................................................270

5.3. ............................................................................................................271 5.3.1. ...................................................................................................271 5.3.2. ..............................................................................................274

5.4. , ................................................................276 5.4.1. .......................................277

- - ................................................... 277 - ......................................................................................... 279 - ...................................................... 281

5.4.2. .............................................................................................283 - ............................................................................................. 284 - - ............................................................................................... 284 - .......................................................................................................... 285 - ........................................................................................... 288 - ...................................................................................................... 290 - ................................................................ 293 - ........................................................... 293 - - ....................................................................................... 294 - - ............................................... 297

5.4.3. ...............................................................................................................................297 - ............................................................ 297 - .............................................................................................. 300

5.4.4. . .............................................300 5.4.5. .............................................................................................................303 5.4.6. ...........................................................................................................................306

- ............................................................................................................. 307 - ............................................................................ 311 - ...................................................................................................... 312

5.5. . .........................................312 5.5.1. . ...................................................313 5.5.2. .........................................................................................314 5.5.3. .................................................................................................317 5.5.4. ..................................................................................................318 5.5.5. ..............................................................................................320 5.5.6. ..................................................................................322 5.5.7. .....................................................324 5.5.8. ......................................325

5.6. .............................................................................................326 5.6.1. ........................................................................................326 5.6.2. .........................................328 5.6.3. . .....................................330

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5.6.4. k- ........................................................................333 5.7. .........................................................................334

5.7.1. ....................................................................................334 - ....................................................................................................... 334 - ............................................................................................................ 338 - ........................................................................... 340

5.7.2. ..............................................................................................340 - .................................................................................. 341

5.7.3. ............................................................................................343 5.7.4. ..................................................................................................................................346 5.7.5. , .......................................................................................348

- p- p- .......................................................................................................... 350 5.7.6. . ................................................353

5.8. ..............................................................................................354 5.8.1. . .................................................................354

- .................................................................. 355 - ................................................................... 355

5.8.2. ..................................................................................................355 5.9. ...............................................................................................................357

5.9.1. ........................................................................................................357 5.9.2. ..................................................................................................................363

6 . NP- ..........................................369 6.1. .............................................................................................369

6.1.1. .....................................................................................................369 6.1.2. ....................................................................................................370 6.1.3. ..........................................................................................................370 6.1.4. ..........................................................................................................................370

6.2. NP- ................................................................................................................373 6.3. .............................................................................................................374

6.3.1. ......................................................................376 6.3.2. .....................................................................................................380 6.3.3. - ....................................................................383 6.3.4. ...........................................................................................................385 6.3.5. ............................................................................................388 6.3.6. ...........................................................................391 6.3.7. ................................................................................................395

6.4. ...........................................................................398 6.4.1. ( ).............................................................398

6.5. ......................................................................................401 6.5.1. "X"- "O".....................................................................................................402 6.5.2. .........................................................................405 6.5.3. - ....................................................................................................406 6.5.4. - ...................................................408

6.6. ...............................................................................................................409 6.6.1. ........................................................................................................409 6.6.2. ..................................................................................................................413

- NP- ................................................................................................................ 413 - ........................................................................................................... 425

7 .......................................................................................427

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7.1. K- ......................................................................427 7.2. .............................................................................................................................434 7.3. .......................................................................................445 7.4. ....................................................................................................450 7.5. ................................................................................................456 7.6. .........................................458 7.7. .................................................................................461 7.8. .............................................................................................464 7.9. ..........................................................................................466 7.10. ......................................................471 7.11. .......................................................................................................474 7.12. .............................................................................................................475

7.12.1. ......................................................................................................475 7.12.2. ................................................................................................................478

8 .........................................................................481 8.1. .............................................................................................................................481 8.2. ...........................................................................484

8.2.1. .....................................................................................................484 8.2.2. ..........................................................................................................495 8.2.3. ..............................................................................................497 8.2.4. . ...............................................504 8.2.5. ............................................................508 8.2.6. - ........................................................................................513 8.2.7. - ........................................................................517 8.2.8. ...................................................................................522 8.2.9. .................................................................................................526

8.3. ..............................................................................................528 8.3.1. .......................................................................................................528 8.3.2. .................................................................................................532 8.3.3. .............................................................................................................533 8.3.4. .................................................537 8.3.5. .....................................................539 8.3.6. .................................................................................540 8.3.7. ..........................................................................................542 8.3.8. ...........................................................543 8.3.9. ............................................................................................................548 8.3.10. .................................................................................................549

8.4. .................................................................551 8.4.1. .............................................................................................................552 8.4.2. ...............................................................................................................553 8.4.3. .......................................................................................................555 8.4.4. ....................................................559 8.4.5. ............................................................................................561 8.4.6. ...................................................................................................564 8.4.7. ............................................................................................565 8.4.8. .............................................................................................................568 8.4.9. ....................................................................................................571 8.4.10. .........................................................................................574

10

8.4.11. .............................................................................................................576 8.4.12. - ............................................................................................................579 8.4.13. ..............................................................................................582

8.5. ...............................................................................................................584 8.5.1. ........................................................................................................584 8.5.2. ..................................................................................................................591

9 ..................................595 9.1. ...............................................................................................................595

9.1.1. ...........................................................................................................596 9.1.2. ........................................................................598 9.1.3. .................................................................601 9.1.4. ...................................................................................604 9.1.5. .......................................................................................605 9.1.6. .....................................................................................607 9.1.7. ...............................................................................................608 9.1.8. . . ...........................................................610

9.2. .......................................................................................................616 9.2.1. ......................................................................617

- .......................................................................................... 618 9.2.2. ..........................................................................621 9.2.3. ..................................................................................................622

9.3. ........................................................................627 9.3.1. .........................................................................................628

9.4. ...............................................................................................................629 9.4.1. ........................................................................................................629 9.4.2. ..................................................................................................................633

10 ..............................................................................................637 10.1. ............................................................................................................................637 10.2. ........................................................................................................638 10.3. ................................................................................640

10.3.1. ...............................................................................................640 10.3.2. ................................................................................641 10.3.3. ........................................................................................643 10.3.4. ...........646 10.3.5. .................................................................................................646 10.3.6. .....................................................................................................648 10.3.7. .............................................................................................649 10.3.8. . .........................650 10.3.9. ............................................................................653 10.3.10. : PackBits ..............................................................655

10.4. ....................................................................................................656 10.4.1. - ......................................................................................656 10.4.2. ...............................................................................................660 10.4.3. .............................................................................671 10.4.4. K ......................................................................................................672 10.4.5. ............................................................................................673

10.5. ..............................................................................................681 10.5.1. ....................................................................683

11

10.5.2. ......................................................................................................686 10.5.3. : MNP-5...................................................................688

10.6. .......................................................................................................690 10.6.1. .....................................................................................................................691 10.6.2. ...........................................................................................................695 10.6.3. .............................................................................................696 10.6.4. ....................................................................697 10.6.5. LZ77. ..........................................................698 10.6.6. LZSS. ..............................................................................................698 10.6.7. FLZ. LZ77 ......................................................................................699 10.6.8. LZW. ................................................................................700 10.6.9. GIF. CompuServe .....................................................................706 10.6.10. ....................................................................707 10.6.11. ................................................................................708 10.6.12. LZW ..................................................................................................709

10.7. ..............................................................................................710 10.7.1. ..................................................................................710 10.7.2. JPEG ..............................................................................................................................711 10.7.3. . MPEG .......................................................713 10.7.4. ......................................................................................................................715 10.7.5. .......................................................................................716

10.8. .............................................................................................................717 10.8.1. ......................................................................................................717 10.8.2. ................................................................................................................725

...........................................................................................................................727

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http://www.nakov.com/algoplus-bugs/submit-bug.phphttp://www.facebook.com/groups/168112146541301http://www.programirane.org/

0 - , 31

1

" :

, , , ."

~

1.1.

,

-.

,

- , .

1.1.1.

,

, , .

, , ,

,

. -. ,

.

.

-

-

. - .

, , () , -

, .

, , ,

, , ax2 + bx + c = 0 .

- :

A, B, C, ... . , ,

- ( ).

,

A a1, a2, ..., an

:

A = {a1, a2, ..., an}

, ai, i = 1,2,...,n, A aiA,

aiA. -

( A - n),

|A|. n = 0, . -,

( 1.1.1.):

1 - 34

, (a, b, ) ,

, , .

A

a

b

1.1.1. .

, ,

[,-1973]. ,

-.

1.1. A -

B, A B. B ( 1.1.1.). -

, B , A

() B. B.

A B A A B B

B A

(a) () () ()

1.1.1. (), (), () () .

1.2. C A B,

a , aA aB. C = AB.

1.3. C = AB A B C,

, A B.

1.4. C = A\B A B C,

, A, B.

1.5. , .

.

-, .

, ,

. ():

{a, a, b} {a, b, b} {a, b, a} {a, b} {b, a}

1.6. , , -

.

1.7. n- , -

n- ().

1 - 35

n- , .

, (a, b, c) (a, c, b) ..

:

1. A = {1,2,4,5,7} B = {2,3,4,5,6}. :

AB, AB, A\B, B\A.

2. A B , AB = A.

B?

3. A B : AB = (AB) \ (AB).

A = {1,2,4,5,7} B = {2,3,4,5,6}.

4. , A B = B A.

: AB, AB, A\B, B\A, AB?

-

, -

. (

3 "" : , ),

.

. , -

, , .

1.8. ( N) -

, 0, 1, 2, 3, ....

, .

- :

, .

, N - , .

, n-

.

1.9. Z :

..., 3, 2, 1 ( ), 0 (), 1, 2, 3, ... ( ).

,

[-1995]. - ,

,

.

-

. ANSI C (American National Standards Institute [ANSIC])

( 1.1.1. Borland C DOS), -

, :

|short| |int| |long|

int ( , DOS

2 , Windows 4).

, unsigned, ,

: (unsigned)(-1).

1 - 36

1.1.1a. Borland C DOS.

1.1.1a. ,

.

, . , ( -

) , ( . overflow)

.

1.10. p/q, p q q -

. Q.

1.11. , :

x = n + 0,d1d2d3... ,

n , di 0 9. 0,d1d2d3...

. ,

kk

k

k

k dddnxddd

n10

1

10100101010010

2121

kN, k 0.

, di .

(.. ) . -

1/3 = 0,333333..... 3

. 1/3 = 0,(3) .

- , 1/7 = 0,(142857).

, , ..

p/q (p,qN, q > 0),

:

= 3,1415926535 ....

, -

355/113, ,

( 6 )

. 22/7.

:

, [-1995]. , -

IEEE (Institute of Electrical & Electronics

Engineers). , .

1.1.1. Borland C DOS.

float 3,4.10

38 , ..., 3,4.10

38 32

double 1,7.10308

, ..., 1,7.10308

64

long double 3,4.104932

, ..., 1,1.104932

80

char 128, ..., 127 8

unsigned char 0, ..., 255 8

short int 32768, ..., 32767 16

int 32768, ..., 32767 16

long int 2147483648, .., 2147483647 32

unsigned short int 0, ..., 65535 16

unsigned long int 0, ..., 4294967295 32

1 - 37

1.1.1. Borland C.

,

( . underflow):

( ,

).

: 1/3

.

, , - ( > 0) ,

( ), - ,

0.

:

, , -

.

- .

m n , m 0. q r (0 r <

m) , n = q.m + r. q n/m, r

. r , , m n (n

m) m|n. ( )

/ %. "",

:

q = n / m;

r = n % m;

1.12. (n m) % z = 0, , n m z

n m (mod z).

n. : ( )

n 10. , n . ,

n , , n

:

#include

unsigned n = 4242;

int main(void) {

unsigned digits;

for (digits = 0; n > 0; n /= 10, digits++);

printf(" %u %u\n", n, digits);

return 0;

}

digits.c

:

1. m n, (m,n) : (7,3), (7,3),

(7,3), (7,3), (3, 7), (3,7), (3,7), (3,7).

2. m n (m 0) -

n = q.m + r, 0 r < m, (q, r ). [-1995]

3. .

?

1 - 38

-

a1, a2, ... , an. S = a1 + a2 + ... + an -

:

n

i

iaS1

,

ni

iaS1

,

ni

iaS..1

R(x), i

:

)(: xRi

iaS

C Sn n : unsigned sum(unsigned n)

{ unsigned i, s = 0;

for (i = 1; i

1 - 39

)(: )(:)(: )(:

)2(xSj xRi

ij

xRi xSj

ij aa

)(&&)()(||)()()(

)3(xRxS

i

xRxS

i

xS

i

xR

i aaaa

S(x)||R(x) S(x)&&R(x) -

, S R, i j.

P = 1.a2.a3.....an , ,

:

n

i

iaP1

,

ni

iaP1

ni

iaP..1

a[], n -

, :

int mult(int a[], unsigned n)

{ unsigned i;

int s = 1;

for (i = 0; i < n; i++) s *= a[i];

return s;

}

mult.c

:

1. .

2. (1), (2) (3) .

3. , (1), (2) (3).

4. (1), (2) (3), :

n

mi

i

mni

i

ni

i aaaa ..1....1

, 1 n m

- , , n-

1.13. x , y , -

:

xy = x.x. ... .x

(y )

y < 0, xy = 1 / x

-y.

(x 0):

xy = x

y-1.x

xy = x

y+1/x

xy1+y2 = x

y1.xy2

xy1.y2 = (x

y1)y2

xy y -

:

double power(double x, unsigned y)

{ double res = x;

unsigned i;

for (i = 1; i < y; i++) res *= x;

return res;

1 - 40

}

power.c

- ( 7.5.) , - -

xy.

xn = y (n , n > 1), x n- y

n yx . , y n- y .

n = 2, y y

y . ,

:

q pq

p

xx

- , x y (x > 1). y

y = d,d1d2d3 ... :

kk

k

k

k ddddy

dddd

xxx 101

10100101010010

2121

, xy

, k.

, -

, - . y = bx b 1, b >

0, y > 0 x. y b

logb y. (x > 0, y > 0, b > 0, b 1, c

> 0, c 1):

)4(log

loglog

)3(log.log

)2(loglog)(log

)1(loglog

b

xx

xyx

yxxy

bbx

c

c

b

b

y

b

bbb

x

b

xb

- , 2

log x log2 x. ln x

log x: H e = 2,71828... ( 1.1.6.)

:

1. - , .

2. (1), (2) (3) , .

- .

n, nN ( n!) 1 n:

n

i

inn1

...2.1! ,

0! = 1.

C n! :

unsigned long factoriel(unsigned n)

{ unsigned i;

unsigned long r = 1;

1 - 41

for (i = 2; i

1 - 42

- ,

struct data, :

struct data {

int a;

int b;

...

} A[m][n];

/ -

. - ( 1.1.1.).

11 12 ... 1 n

a 21 22 ... a 2 n

m 1 a m 2 ... a mn

11 12 ... 1 n

a 21 22 ... a 2 n

m 1 a m2 ... a mn

a) ) 1.1.1. : () () .

/* */

for (i = 0; i < m; i++)

for (j = 0; j < n; j++)

scanf("%d", &A[i][j]);

/* */

for (i = 0; i < n; i++)

for (j = 0; j < m; j++)

scanf("%d", &A[j][i]);

/* */

for (i = 0; i < m; i++) {

for (j = 0; j < n; j++)

printf("%.3d", A[i][j]);

printf("\n");

}

matrix.c

Amn Bmn Cmn , cij = aij + bij ( i = 1, 2,...,

m, j = 1, 2, ..., .n), 1.1.1.

11 12 ... 1n

a21 22 ... a2n

...

m1 am2 ... amn

b11 b12 ... b1n

b21 b22 ... b2n

b m1 bm2 ... bmn

Cmn = + =

11+b11 12+ b12 ... 1n+ b1n

a21+ b21 22+ b22 ... a2n+ b2n

m1+bm1 am2+ bm2 ... amn+ bmn

1.1.1. .

for (i = 0; i < m; i++)

1 - 43

for (j = 0; j < n; j++)

C[i][j] = A[i][j] + B[i][j];

summat.c

Amn Bnp Cmp, :

n

k

kjikij bac1

. , i = 1,2,...,m j = 1,2,...,p.

-, m.p.n

. n > m n > p, n3.

for (i = 0; i < m; i++)

for (j = 0; j < p; j++) {

C[i][j] = 0;

for (k = 0; k < n; k++)

C[i][j] += A[i][k] * B[k][j];

}

multmat.c

, , . -

, nlog 7

( n2,81

) , -

. , -

, 7.6., .

:

1. unsigned a[MAX][MAX].

void fillMatrix(unsigned a[][MAX], unsigned n),

a[][] :

0 20 19 17 14

1 0 18 16 13

2 5 0 15 12

3 6 8 0 11

4 7 9 10 0

2. unsigned a[MAX][MAX]. , -

, n = 5 :

1 16 15 14 13

2 17 24 23 12

3 18 25 22 11

4 19 20 21 10

5 6 7 8 9

1.1.2.

: a1, a2, ..., an. -

P = a1.a2..an.

, ai = i, i = 1,2,...,n, P = 1.2 ... n = n!.

1 - 44

,

1.1.1., .

n!. n! ,

-. 10! e 3628800, . 20!

C.

- , ,

, :

P

n

i

ia1

10log1, [x] -

, - x.

, ,

P [1+log10(P)], -

:

log P = log(a1.a2. an) = log a1 + log a2 + + log an.

n!

#include

#include

const unsigned long n = 123;

int main(void)

{

double digits = 0;

unsigned i;

for (i = 1; i

1 - 45

, .

100

, ,

, (, ,

).

, :

[a, b]?

?

n- ?

(x) ,

x, (x) . (

6.2.), (x), ( , ln x loge x):

( ) (x) x / ln(x a),

, - x.

- = 1.

. n- [n.ln(n)]. -

[n(ln(n) + ln(ln n 1))].

. x 1/ln(x).

,

[Primes-1][Primes-2]:

1. n > 2 . 2. n > 17 . 3. -

.

4. n > 5 . 5. .

. n2+m

2 n

2+m

2+1.

. n2+1.

. () n2 (n+1)

2 .

:

1. (x) - , -?

2. .

3. .

-

, , :

[2, 2

p 1] p , , , p .

1 - 46

,

, ,

- .

(). p , (p1)! 1 (mod p).

(p1)!, - -

2

p 1 .

, 2

p 1:

p (). , ,

p x, x > p , , p p = x.y, y < p , ..

- p . :

#include

#include

const unsigned n = 23;

char isPrime(unsigned n) /* 1, , 0 */

{ unsigned i = 2;

if (n == 2) return 1;

while (i

1 - 47

{ unsigned i = 0;

while (i < K && prime[i] * prime[i]

1 - 48

1.1.3. .

: ,

( ).

:

2 n :

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... , n

2. ,

:

(2), 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... , n

-, :

3. , 3:

(2), (3), 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... , n

, 5 5-:

(2), (3), 4, (5), 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... , n

"" , -

i . ,

,

. -

:

1) sieve[] . -, , 1. - i,

(.. -

). i = 2.

2) i, sieve[i] 0. i e . 3) M 1 sieve[k], k = i, 2i, 3i, , (n/i).i (..

i ).

4) i n, 2, . #include

#define MAXN 30000

/* n */

1 - 49


char sieve[MAXN];

void eratosten(unsigned n)

{ unsigned j, i = 2;

while (i

1 - 50

}

void findPrimes(unsigned n)

{ unsigned i = 2;

while (i < n) {

if (isPrime(i)) {

primes[pN] = i;

pN++;

printf("%5u", i);

}

i++;

}

}

int main(void) {

findPrimes(n);

printf("\n");

return 0;

}

rproc.c

, n ,

. [a,b] -

a (a>>1), - -

.

2, 3 5.

:

n = 30.q+r, r[0..29] r[0, 1, 2, ..., 14, 15]. 30.q = 2.3.5.q

30 8. . .

4k/15 , :

30.q1, 30.q7, 30.q11, 30.q13

-

.

:

1. : 3) k = i2, -

i .

. .

2. .

3. , -

4 .

-

( ) P (P>1)

() P1q1. P2

q2. .Pnqn, P1 < P2 <

< Pn P , a qi . [-1995].

:

520 = 23.5

1.13

1

64 = 26

2345 = 51.7

1.67

1

1 - 51

A , -

( 2 1.1.5.), :

P.

1) i = 2. 2) k = 0. P i, k .

3).

3) k > 0, ik.

4)

4) P > 1, i 2).

, . -

:

#include

unsigned n = 435; /* , */

int main(void) {

unsigned how, i, j;

printf("%u = ", n);

i = 1;

while (n != 1) {

i++;

how = 0;

while (0 == n % i) {

how++;

n = n / i;

}

for (j = 0; j < how; j++)

printf("%u ", i);

}

printf("\n");

return 0;

}

numdev.c

:

.

- ,

: a1, a2, ..., an. ,

P = a1.2. ... .an.

1.1.2., -

. ,

: , , 2 5,

, 2, , 5.

, , :

1) i (i = 1, 2,..., n) ai ai = 2Mi

.5Ni

.bi, bi % 2 0, bi % 5

0.

2) P = cnii

ni

i NM

.5.2 ..1..1 , (c ), a

n

i

Mi1

n

i

Ni1

.

, :

25, 4, 20, 11, 13, 15

1 - 52

:

20.5

2.1, 2

2.5

0.1, 2

2.5

1.1, 2

0.5

0.11, 2

0.5

0.13, 2

0.5

1.3,

4 . :

25.4.20.11.13.15 = 4290000.

- .

.

n!.

][log

1

5

5

n

kk

n . - ,

2 5 n . #include


int main(void) {

unsigned zeroes = 0, p = 5;

while (n >= p) {

zeroes += n / p;

p *= 5;

}

printf(" %u! %u\n", n, zeroes);

return 0;

}

factzero.c

:

1. ,

.

2. , -

.

3. , n!

1.1.4.

-

1.15. ,

2p1, p e .

39 p, 2p1 :

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941,

11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269,

2976221, 3021377, 6972593, 13466917

37 p 37

. ( 1999 2001 .),

38- 39-, ,

-. p. .. - .

: , -

. ,

. 23- 1963 .,

,

1 - 53

"211213

1 " (

1.1.4.).

- , , :

213466917

1 8107892 (??39- M )

26972593

1 2098960 (??38- M )

23021377

1 909526 (37- M )

22976221

1 895932 (36- M )

21398269

1 420921 (35- M )

- , -

. ,

(

-).

- : , ""

38- , $50000.

$250000 [Primes-3].

1.1.4. 2

11213-1 .

:

p = 2, 3, 5, 7, 11, = 2p 1 .

, .

1870 , -

- ,

( e -

[Guinier-1991]).

(LucasLehmer Test, 1930) :

:

1 = 4

n+1 = (En )2 2

:

4, 14, 194, 37634, .

(-). m = 2p 1 ( p )

, :

(Ep1) % (2p 1) = 0

-

.

.

:

1. n ,

-.

2. n ,

-. .

3. 2n1 , n ?

1 - 54

- .

1.16. n ,

( n ).

3 :

6 = 1 + 2 + 3,

28 = 1 + 2 + 4 + 7 + 14,

496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.

:

8128, 33550336, 8589869056,

, ( n = 1,

2, 3, ), 5-

. :

. () 2p1 , 2

p1.(2

p1) .

n -

: pi n

( 10 2, 3, 5, 7, 13, 17, 19, 31, 61, 89). ,

n,

. ,

: 2

. number[]:

.

: number[k-1] ( k ),

number[k-2] .. k- number[0].

, number[], :

i = 0; number[i]++;

while (10 == number[i]) { number[i] = 0; number[++i]++; }

if (i == k) k++;

2,

, .. 9, -

, while(10 == number[i]), .

, :

unsigned i, carry = 0, temp;

for (i = 0; i < k; i++) {

temp = number[i] * 2 + carry;

number[i] = temp % 10;

carry = temp / 10;

}

if (carry > 0) number[k++] = carry;

, 10 , -

mPrimes[]:

#include

#define MN 10

unsigned mPrimes[MN] = { 2, 3, 5, 7, 13, 17, 19, 31, 61, 89 };

unsigned k, number[200];

void doubleN(void)

{ unsigned i, carry = 0, temp;

1 - 55

for (i = 0; i < k; i++) {

temp = number[i] * 2 + carry;

number[i] = temp % 10;

carry = temp / 10;

}

if (carry > 0)

number[k++] = carry;

}

void print(void)

{

unsigned i;

for (i = k; i > 0; i--)

printf("%u", number[i-1]);

printf("\n");

}

void perfect(unsigned s, unsigned m)

{

unsigned i;

k = 1; number[0] = 1;

for (i = 0; i < m; i++)

doubleN(); /* 2^i */

number[0]--; /* {2,4,8,6} */

for (i = 0; i < m - 1; i++)

doubleN();

printf("%2u-o = ", s);

print(); /* */

}

int main(void) {

unsigned i;

for (i = 1; i

1 - 56

2. , ,

( 1 ) 2. ?

1.1.5. , .

1.17. n- .

k-

:

)(1)...1(

)1)...(1.(

kk

knnnC

k

n

n

k

, n

kC

k

n -

, n,

. .

n , (*) :

(**))!(!

!

knk

nC

k

n

n

k

, ,

n k ( 1.3.3.) (?).

, (a+b)n, -

:

n

i

iinnnnn bai

nba

n

nba

nba

nba

0

0110 ...10

, [Knuth-

1/1968]. .

1.1.5. n

kC , 0 k n < 10.

n

0

n

1

n

2

n

3

n

4

n

5

n

6

n

7

n

8

n

9

n

0 1 0 0 0 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0 0 0

2 1 2 1 0 0 0 0 0 0 0

3 1 3 3 1 0 0 0 0 0 0

4 1 4 6 4 1 0 0 0 0 0

5 1 5 10 10 5 1 0 0 0 0

6 1 6 15 20 15 6 1 0 0 0

7 1 7 21 35 35 21 7 1 0 0

8 1 8 28 56 70 56 28 8 1 0

9 1 9 36 84 126 126 84 36 9 1

1.1.5. .

, --

1.1.5., :

1 - 57

10

n

nn (1)

kn

n

k

n (2)

1

11

k

n

k

n

k

n (3)

1.1.5. - ,

:

n = 0 1

n = 1 1 1

n = 2 1 2 1

n = 3 1 3 3 1

n = 4 1 4 6 4 1

n = 5 1 5 10 10 5 1

...

1.1.5. .

1 ( (1) (2)),

( (3)). n-

, (n1)- .

k

nC

n

k

, n- k-

n

kC . , i

i1, .

:

lastLine[], n k n

kC

.

#include

/* */

#define MAXN 1000


const unsigned k = 3;

unsigned long lastLine[MAXN + 1];

int main(void) {

unsigned i, j;

lastLine[0] = 1;

for (i = 1; i = 1; j--)

lastLine[j] += lastLine[j - 1];

}

printf("C(%u,%u) = %lu\n", n, k, lastLine[k]);

return 0;

}

pascalt.c

1 - 58

, (*)

.

(**) n! k!(nk)!,

, : n = 100 k =

2 100! 2!.98! ( 150 ), 100

2C 4950.

2 n

kC ( )

1) 7

3C . (*) -

, .

: 7! = 1.2.3...7 =

1.2.3.22.5.(2.3).7 = 2

4.3

2.5

1.7

1.

2) : 3!.(73)! = 1.2.3. 1.2.3.22 = 2

4.3

2.

3) : 1

.75

.32

.7.5.32 11

24

1124

.

4) : 5

1.7

1 = 35.

2.

#include

#define MAXN 500

unsigned long n = 7;

unsigned long k = 3;

unsigned long pN, primes[MAXN], counts[MAXN];

void modify(unsigned long x, unsigned how)

{ unsigned i;

for (i = 0; i < pN; i++)

if (x == primes[i]) {

counts[i] += how;

return;

}

counts[pN] = how;

primes[pN++] = x;

}

void solve(unsigned long start, unsigned long end, unsigned long inc)

{ unsigned long prime, mul, how, i;

for (i = start; i 0)

modify(prime, inc * how);

prime++;

}

}

}

unsigned long calc(void)

{ unsigned i, j;

1 - 59

unsigned long result = 1;

for (i = 0; i < pN; i++)

for (j = 0; j < counts[i]; j++)

result *= primes[i];

return result;

}

int main(void) {

printf("C(%lu,%lu)= ", n, k);

pN = 0;

if (n - k < k) k = n - k;

solve(n - k + 1, n, 1); /* (nk+1),...,n */

solve(1, k, -1); /* 1,...,k */

printf("%lu\n", calc());

return 0;

}

cnk.c

( , ..

)

n

kC , , ,

.

: "" -

:

kn

n

k

n

, .

:

1. (1), (2) (3), -

. , (1) :

1!0!

!

)!(!

!

1!!0

!

)!0(!0

!

0

0

n

n

nnn

nC

n

n

n

n

n

nC

n

n

n

n

2. (1), (2) (3).

3. 1 (pascalt.c) n

kC

lastLine[] n+1 , - n,

. : ,

, 1 i, 1 k,

- . , k

n, (2) n

knC , n

kC .

1.1.6.

-

.

, - -

1 - 60

.

0, 1, 2, 3, 4, 5, 6, 7, 8 9, (

,

- , .).

, , .

( 10 ). -

, ( 0 1) ( -

) . , p p < 0,

.

, -

.

0 1. 11(10) 1011(2) (

, .).

16 . ,

0 9 , A, B, C, D, E

F, 10, 11, 12, 13, 14 15. 254(10)

FE(16). , -

e = 2,718281828... ( .

an

n

n

1

1 n). -

.

e (

),

3 2.

, 3 - e.

. -

0 0 V, 1 5 V. -

. -

, , ,

1960 . , 1, 0 1. , 16 2 (16=2

4).

0 15 4 ( 0 1). --

8 .

, 0 255, .. 256 .

256 = 162,

.

- (4 ),

,

. - ,

10101000110111(2) . :

10 | 1010 | 0011 | 0111

: 0010 | 1010 | 0011 | 0111

2A37(16). ,

( ). -

- . ,

. , 153 ,

1 - 61

6. , 1, 5 3, ,

3, 3 , -

0, 1 2.

- , -, . , .. -

, . -

123 3 3, 34 30. -

( 1.1.7.),

. - : -

. -. -

, -

p, p . -

anan-1...a0 (p), i (1 i n) .

:

A = anpn + an-1p

n-1 + ... a1p + a0

,

-.

:

1. 0 1?

2. 17 17 : 2; 8; 16.

3. 17 17 2; 8; 16.

4. ,

: 111, 110100, 1110100101, 10010101, 10101010101 10111110101.

5. , -

: 11, 11001, 1010101, 111111, 1010101000, 10101101000 11010111000.

6. ?

7. 17 17 .

- p-

,

p- : A p

, A 0, ,

(?). 29

( ):

29:2=14:2=7:2=3:2=1:2=0

1 0 1 1 1

: 29(10) = 11101(2). convert()

: char getChar(char n) /* , n */

{ return (n < 10) ? n + '0' : n + 'A' - 10; }

void reverse(char *pch)

{ char *pEnd;

for (pEnd = pch + strlen(pch) - 1; pch < pEnd; pch++, pEnd--) {

char c = *pch;

*pch = *pEnd;

*pEnd = c;

1 - 62

}

}

void convert(char *rslt, unsigned long n, unsigned char base)

/* n (n >= 0) */

/* base */

{ char *saveRslt = rslt;

while (n > 0) {

*rslt++ = getChar((char)(n % base));

n /= base;

}

*rslt = '\0';

reverse(saveRslt);

}

base.c

- 1, ,

p.

A p- : -

A p, . -

0,125 :

0,125.2=0,25.2=0,5.2=1,0

0,125(10) = 0,001(2). A

p- () . -

.

convertLessThan1() A (0 < 1) cnt

, . void convertLessThan1(char *rslt,

double n,

unsigned char base,

unsigned char cnt)

/* 0 1 0 <

1, ,

, . convertReal(): void convertReal(char *rslt, double n,

unsigned char base, unsigned char cnt)

/* n base */

{ double integer, fraction;

1 - 63

/* */

if (n < 0) {

*rslt++ = '-';

n = -n;

}

/* */

fraction = modf(n, &integer);

/* */

convert(rslt, (unsigned long)integer, base);

/* ( ...) */

if ('\0' == *rslt) *rslt++ = '0';

else rslt += strlen(rslt);

*rslt++ = '.';

/* */

convertLessThan1(rslt, fraction, base, cnt);

if ('\0' == *rslt) {

*rslt++ = '0';

*rslt = '\0';

}

}

base.c

:

1. 157 : 3;5;7;14.

2. 0,321 : 3;5;7;14.

3. 157,321 : 3;5;7;14.

4. -

p- .

5.

p- .

- p- .

p- -

A,

. , , 12734(8), -

: 12734(8) = 1.8

4 + 2.83 + 7.82 + 3.8 + 4 = 5596(10)

, 10 . -

, 8,

. - ,

n- n

. (1) (2).

(1) Pn(x) = a0xn+a1x

n-1+...+an-1x+an

(2) Pn(x) = an+x(an-1+x(an-2+...+x(a2+x(a1+xa0))...))

calculate():

1 - 64

char getValue(char c) /* c */

{ return (c >= '0' && c = 0 */

{ unsigned long result;

for (result = 0; '\0' != *numb; num

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