Number Theory

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2006

, , 1998-1999 3 . . , , , . . , , . . - . , , . . 14+14 1 , , , , . , , Gauss . . . . , , , . . : , . ( , ) . : . . . . . , , ! 22 2006 ..

1

14 14

O . / .

. :[1.] J. HUNTER , 1974 [2.] ADOLPH HURWITZ ( ) .. , 1981 [3.] , , 1997 [4.] .. , . , . 2000

. - :[5.] N. . - 2006 http://server.math.uoc.gr/~tzanakis/Courses/NumberTheory/textbook_main.pdf [6.] . 17 18 . - 1999 http://server.math.uoc.gr/~antoniad/hist.ps

. : . : [7.] G. H. HARDY, E. . WRIGHT An Introduction to the Theory of Numbers OXFORD, 1959 [8.] EDMUND LANDAU Elementary Number Theory ( : Jacob E. Goodman) CHELSEA, 1966

. - :[9.] LEO MOSER An Introduction to the Theory of Numbers The Trillia Lectures on Mathematics , 2004 (1957) http://www.trillia.com/distr4/moser-number-a4-one.pdf [10.] DAVID A. SANTOS Number Theory for Mathematical Contests 2005 REVISION http://www.openmathtext.org/lecture_notes/number_theory_book.pdf [11.] EDWIN W.CLARK Elementary Number Theory Department of Mathematics. University of South Florida, 2003 http://www.math.usf.edu/~eclark/elem_num_th_book.ps [12.] A. BAKER Algebra & Number Theory Department of Mathematics, University of Glasgow., 2003 http://www.maths.gla.ac.uk/~ajb/dvi-ps/3q-notes.pdf [13.] ROBIN CHAPMAN A Guide to Arithmetic 1994 http://www.maths.ex.ac.uk/~rjc/notes/arith.pdf [14.] FELIX LAZEBNIK Elements of number theory: Lecture Notes 2005 http://www.math.udel.edu/~lazebnik/papers/ElementsThNum.pdf

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1.1. 0, 1, 2, 3, 4, 5, .... N . N ={0, 1, 2, 3, ...}. . . +1. 2 3, 5 6. 0 : 56 55, 13 12. . 0 . . . N *. N * ={0, 1, 2, 3, ...}. ( ) : , , , , k, m, n. , , . .

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1.2.

1 1 1 1 + + +...+ 2 2 5 10 +11 + p + p 2 + p 3 +...+ p n

1 + 1 2 + 1 2 3 +...+1 2 3 ...n1 1 1 1 + + +...+ k 2 3 5 2 +1

1.3.

(1 + 1)(1 + x )(1 + x 2 )...(1 + x )

(1 + 1)(1 + x )(1 + x 2 )(1 + x 3 )...(1 + x 2 ) (1 + 1)(1 + x 2 )(1 + x 4 )...(1 + x 2 )

1.4. 1.5. 1.6.

3x-2 31. x; 81;

0, 1, 2, 3, .... Z Z ={0, 1, 2, 3, ....}. Z * Z * ={ 1, 2, 3, ....}. . :

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1.7. 1.8. 1.9.

[-1,5]; x; [-480, 1999];

1.10.

;

1.11. 1.12.

81;

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2 13 5 2 , , , 3 18 1 3

5 , 1 (

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Q=

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U V W

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2 4 = . 3 6 , , , ,

, . .

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. : . .

13. (0, 1). 14. x x x +11 , ; 3 12

15. . . N Z

Z Q .

Q 2 . 3

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16. x x+12 ; 17. x o 18 ; 11x x

18.

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( =) . ( ) . ( ) . ( ) . { } ( ) .. ={0, 1, 2, 3}, ={0, 1, 2, 3, 4, 5} . ={0, 1, 2, 3}, ={0, -1, 2, -3, 4, -5} ={0, 2} ={0, 1, -1, 2, 3, -3, 4, -5}

B4. 5.

A

A2.

0. 1. 3.1.

3.

B0. 1. 3. 4.

AB 2.

AB

1.19. A ( 2 ,7) ( 5 , 7 )

1.20. .. www.nsmavrogiannis.gr

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M ( ) A [-1, 5) (-2, 4]

1.21. , .. 4-7 4 7 . .

; + . - : (0)

N

Z

Q

1.22. , . :

(+)(-) ,

+ + 2 + 2 || , , , , -

1.23. 90, 100, 120, 130 2 1, Z ;

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( : ).

1.24. 90, 100, 120, 130 20+30, Z ; 1.25. 90, 100, 120, 130 202+30, Z ; 1.26. 90, 100, 120, 130 2+2+1, ,Z ; 1.27. 22 ,( 22 )n (n ) ; 1.28. ;n

1.29. , , ; (+1)2+(+1)2 (+1)3+(+1)2 ++2

1.30. , + ,Z. , + 2+3, -5+6, 12+0, 0+(-1) , . 2+ , ()+(). 5 2 8. (;). , , : . .. (-2)2+13=(10)2+(-7)3.

1.31. :

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I. x, y , x-y x-y , . , , . II. x , x , . , , . III. x , , x , .

1.32. , ; . . ==0 0 , . , 0 , x, y x+y= x+by=. x+by= . . , x+by= . 2x+3y=5 (1,1) (-2, 3). x, y 2x+4y=5 ( ). 5 .. www.nsmavrogiannis.gr

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2 4. 6 (12+14=6). 2x+4y=5 2x+4y=6 .

2. 2.1. . . . .

2.2. Euler 2++41( Euler) ( 1 : ). Euler 1, 2, 3, ..., 39

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

++41 43 47 53 61 71 83 97 113 131 151 173 197 223 251 281 313 347 383 421 461

2

21 22 23 24 25 26 27 28 29

++41 503 547 593 641 691 743 797 853 911

2

30 31 32 33 34 35 36 37 38 39

971 1033 1097 1163 1231 1301 1373 1447 1523 1601

=40 : 402+40+41=40(40+1)+41=40.41+41=41.(40+1)=412 41 .

2.3. . : 30 .

x-1 +1 -1. : 1 2 3 4 5 x-1 x1-1 x2-1 x3-1 x4-1 x5-1 x-1 . x-1 (x-1)(x+1) (x-1)(x2+x+1) (x-1)(x+1)(x2+1) (x-1)(x4+x3+x2+x+1)

. 1941 =105 .. www.nsmavrogiannis.gr

M ( ) x105-1=(x-1) (x6+x5+x4+x3+x2+x+1)(x4+x3+x2+x+1) (1-x+x5-x6+x7-x8+x10-x11+x12-x13+x14-x16+x17-x18+x19-x23+x24) (x2+x+1)(1-x+x3-x4+x6-x8+x9-x11+x12) (1-x+x3-x4+x5-x7+x8) (1+x-x43+x46+x47+x34+x35+x36-x39-x40-2x41-x42-x22-x26-x28+x31+x32+x33-x20+x15+x48x6-x5+x2-2x7-x24+x12-x8+x13+x14+x16+x17-x9)

11

2. x-1 1 104.

2.4. Goldbach. 4 ( 2) 6 3+3 8 5+3 10 5+5 12 7+5 14 7+7 11+3 16 5+11 18 15+3 Goldbach . . . .

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Goldbach 1 .

2.5.

1+2+3+...+; . . 1=1 1+2=3 1+2+3=(1+2)+3=3+3=6 1+2+3+4=(1+2+3)+4=6+4=10 1+2+3+4+5=(1+2+3+4)+5=10+5=15 1+2+3+4+5+6=(1+2+3+4+5)+6=15+6=21 1+2+3+4+5+6+7=(1+2+3+4+5+6)+7=21+6=27 ................................................................................ ................................................................................ 1, 3, 6, 10, 15, 21, 27 ... ; : . 1, 2, 3, 4, ... , . 1+2+3+...+. ( =7)

.

1

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1 .

1 2 2 1 . 2 1 1 1 1 1+2+3+...+= 2 + = ( 2 + )= ( +1) 2 2 2 2

1 + 2 + 3 + ... + =

( + 1)2

. 1=1=1.2/2 1+2=3=1.1/2 1+2+3=6=3.4/2 1+2+3+4=10=4.5/2 1+2+3+4+5=15=5.6/2 1+2+3+4+5+6=21=6.7/2 1+2+3+4+5+6+7=27=7.8/2 ................................................................................ ................................................................................ . 1+2+3+...+=(+1)/2. .

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7 .

1 ( ) 1+2+3+...+7= 7(7 + 1) . 2 =8.

1 + 2 + 3 + ... + 7 + 8 = (1 + 2 + 3 + ... + 7) + 8 = 1 1 9 1 1 7 8 + 8 = 7 8 + 8 = 7 + 18 = 8 = 8(8 + 1) . 2 2 2 2 2 =8. ; . =k k.

1 k 1+2+3+...+k= k(k + 1) 2 =k+1. :1 + 2 + 3 + ... + k + (k + 1) = (1 + 2 + 3 + ... + k ) + (k + 1) =

=

1 1 1 k(k + 1) + (k + 1) = k + 1(k + 1) = (k + 1)(k + 2) 2 2 2

1 =k =k+1. . . .

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2.6. : .

A 1. =k =k+1.

.

2.7.

2

0 > 1 : (1 + ) > 1 + . ( Bernoulli)

2.8. Bernoulli