Zahlentheorie für Einsteiger, 6. Auflage

Andreas Bartholomé | Josef Rung | Hans Kern Zahlentheorie für Einsteiger Aus dem Programm Mathematik für Einsteiger ...

Author: Andreas Bartholomé | Josef Rung | Hans Kern

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Andreas Bartholomé | Josef Rung | Hans Kern Zahlentheorie für Einsteiger

Aus dem Programm

Mathematik für Einsteiger

Algebra für Einsteiger von Jörg Bewersdorff Algorithmik für Einsteiger von Armin P. Barth Diskrete Mathematik für Einsteiger von Albrecht Beutelspacher und Marc-Alexander Zschiegner Finanzmathematik für Einsteiger von Moritz Adelmeyer und Elke Warmuth Graphen für Einsteiger von Manfred Nitzsche Knotentheorie für Einsteiger von Charles Livingston Stochastik für Einsteiger von Norbert Henze Strategische Spiele für Einsteiger von Alexander Mehlmann Zahlen für Einsteiger von Jürg Kramer Zahlentheorie für Einsteiger von Andreas Bartholomé, Josef Rung und Hans Kern

www.viewegteubner.de

Andreas Bartholomé | Josef Rung | Hans Kern

Zahlentheorie für Einsteiger Eine Einführung für Schüler, Lehrer, Studierende und andere Interessierte Mit einem Geleitwort von Jürgen Neukirch 6., überarbeitete und erweiterte Auflage STUDIUM

Bibliografische Information Der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über abrufbar.

Dr. Andreas Bartholomé und Josef Rung unterrichten am Hans-Leinberger-Gymnasium in Landshut. Anschrift: Jürgen-Schumann-Straße 20, 84034 Landshut Dr. Hans Kern unterrichtet am Schyren-Gymnasium in Pfaffenhofen/Ilm. Anschrift: Niederscheyerer Straße 4, 85276 Pfaffenhofen Online-Service: http://www.andreasbartholome.de

1. Auflage 1995 2., überarbeitete Auflage 1996 3., verbesserte Auflage 2001 4., durchgesehene Auflage 2003 5., verbesserte Auflage 2006 6., überarbeitete und erweiterte Auflage 2008 Alle Rechte vorbehalten © Vieweg+Teubner Verlag |GWV Fachverlage GmbH, Wiesbaden 2008 Lektorat: Ulrike Schmickler-Hirzebruch | Susanne Jahnel Der Vieweg+Teubner Verlag ist ein Unternehmen von Springer Science+Business Media. www.viewegteubner.de Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlags unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, dass solche Namen im Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei zu betrachten wären und daher von jedermann benutzt werden dürften. Umschlaggestaltung: KünkelLopka Medienentwicklung, Heidelberg Druck und buchbinderische Verarbeitung: MercedesDruck, Berlin Gedruckt auf säurefreiem und chlorfrei gebleichtem Papier. Printed in Germany ISBN 978-3-8348-0440-2

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' ! 0 3 4 r(0) := a, r(1) := b 2 r(n) = 0 r(n−1) "# 0 3 r(n + 1) 4 r(n − 1) = q(n − 1) · r(n) + r(n + 1) - 0 % {r(n) = 0|n ∈ N} & ' r(n + 1) 0 &4

r(0) = a = q(0) · r(1) + r(2) = q(0) · b + r(2) r(1) = q(1) · r(2) + r(3) = r(n) = q(n) · r(n + 1) 5% i ∈ {0, 1 . . . , n} ggT(r(i), r(i + 1)) = ggT(a, b) r(n + 1) = ggT(a, b). 6 4 • •

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1049 = ggT(955354721, 330435) a b = 0 b a mod b ! " # $ % & $' ($ )* + , % - ) " % - . /0 $ 1 2 3 4 5! $ 4 1! ! ! ! 64 7 8 % 1 9 % " ! 4 ! ! 5. : ; &)

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rk = (a·x2 + b·y2 )− qk (a·x1 + b·y1 ) = a·(x2 − qk x1 )+ b·(y2 − qk ·y1 ) ∈ aZ + bZ. + ( ' ggT(a, b) aZ + bZ !! % aZ + bZ = cZ c = ggT(a, b)

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d ggT(a, b) 1 = ggT(x, y).

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a = dx

b = dy

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Dividend = xalt · a + yalt · b; Divisor = xneu · a + yneu · b Dividend = q · Divisor + Rest. Rest = (xalt − q · xneu) · a + (yalt − q · yneu) · b. ! . xneu := xalt − xmitte · q 4 yneu := yalt − ymitte · q; xalt := xmitte4 yalt := ymitte; xmitte := xneu4 ymitte := yneu; aalt := amitte4 amitte := aneu; ! $ &' . &' $ ! $ #

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f (n) f (2n) f (4n) f (6n), . . . ggT

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U = 12 Z + 13 Z ⊂ Q Q

Q R

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a +m (b +m c) = r(a + r(b + c)) = r(a + (b + c)) = r((a + b) + c) = r(r(a + b) + c) = (a +m b) +m c. # a +m 0 = r(a + 0) = r(a). $% a = 0 ∈ Z/mZ 0 +m 0 = r(0) = 0. $% a ∈ {1, . . . , m − 1}

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2

Z/mZ , " " - " ". / - 0 1139225 # 7 - " # 1 1139225 ≡ 325 = (36 )4 · 3 = 3 mod 7 11392 , ." 0 # 7 - # 1139225 ≡ 3 . . . 2# Z/mZ ,. * / " - "" , 2 - %" 3 4 / !" - " Z/mZ ,

0"

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r : N n → n mod m ∈ Z/mZ #

# " $ % r(a + b) = r(a) +m r(b) $% r(a · b) = r(a) ·m r(b) # a, b ∈ N0

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a ∈ R a = 0 a α α ! α = β · γ β, γ ∈ R d(α) ≥ d(β) q, r ∈ R β = q · α + r β " α # β r $% r = 0# % d(β) ≤ d(r) < d(α) # & α "

a

' r = 0 ' β = q · α = q · β · γ 2 β

α (

p > 2

5 p p Z[φ] = R p α α Z[φ]

5 ) * p # Z/pZ a a2 − a − 1 = 0 Z/pZ ! ρ : Z[φ] x + yφ → x + ya ∈ Z/pZ

* $ ρ(a − φ) = a − a = 0 Z/pZ a − φ ∈ +(ρ) ! a − φ ∈ / pR α ∈ R

pR +(ρ) = αR α " p β ∈ R p = α · β p2 = d(p) = d(α) · d(β)# α, β d(α) = p 2 , ( # ( Z , * (

p > 2 α

Z[φ]

p

! p = N (α) = x2 +xy −y 2 x p % y p p p2 ! Z/pZ . x2 + xy − y 2 −x2 y 2 y − −1=0 x x ! Z/pZ . x2 − x − 1 = 0 / α

(2α − 1)2 = 4(α2 − 4α − 4) + 5 = 5 5 ) * p 0 1 2 α, β # p = α · β

p2 = N (α) · N (β) N (α) = p 0 N (α) = 1

N (α) = p2

α

β 2 p

!

" #$ % & ' # % ( ! ) *

+$ , % -

' - & a = 0 % . ' . / a % % 0 d(a) a .

# - #

a . # π a = π · b #$

b ∈ R d(b) < d(a) b . ' . b = π1 · · · πn % a = π · π1 · · · πn . #

N 2 , ! 1

(!

- % ' (% 2(% 3 4 5 67 % ' - , . # 8

% & % ( , % . 0 ' '

. # Z[φ] % ,

$ %

)

0 ' 0% ( . Z[φ]

α := a+ bφ ∈ Z[φ] a + b(1 − φ)

Z[φ] ρ(α) =

1 9 # #$ % 0% ( %

π ∈ Z[φ] N (π) = ±1·p N (π) = ±1·p2 p ∈ N

N (π) = p · a a ∈ N π . π ) p a

p = π · α α ∈ Z[φ] N (π) = π · ρ(π) = π · α · a ρ(π) = α · a ρ(π) Z[φ] α a a N (a) = a2 = 1 a = ±1 N (π) = ±p α Z[φ] p2 = N (p) = N (π) · N (α) = ±1 · N (π)

π a π · ρ(π) = p · π · α α ∈ Z[φ] ρ(π) = pα α N (ρ(π)) = N (π) = p2 · (±1) 2 !" # $

# % & % Z[φ] ' "

n α ∈ Z[φ] N Z[φ] Z[φ] " p " 5 ( ) * p +,- # " $ 5n ± 2 !" . " /# $ n = ±1 / , 0 % / , + " 0 ( + n = N (α) " α ( Z[φ] α = N (π1 · · · πk ) n = p · a " p % a . -

n = p · a = N (π1 · · · πk ). 1 -( (2 # p % N (π1 ) p = N (π1 ) a & Z[φ] $ ( % a 3 / , + " " $ p2 = N (π1 ) (2 # 4 p2 + " #0 2 √ √ Z[i] Z[ 2] Z[ 3]

α !" # x2 +x−3 = 0 $ %& '() *+ ,- . / 0 1$ 2 3 4 56 2 Z[α]

√

Z[φ] = { a2 + 2b 5|a, b ∈ Z ! ! } x+yφ ∈ √ Z[φ] " x + yφ = a2 + 2b 5 ! N (x + yφ) = x2 + xy − y 2 = 1 2 5 2 4a − 4b √

# $% & Z[ −5] $ 21 ' ()

! ! * ! + ' +

35 !" # $ % & ' ( " ) * # $ %

& ' ( ) * # #

" + ## #$ #% #& #' #( #)

! + , #* $ $ $# $$ $% $& &-

. /+0 1 + " + 2 /

3 4 " + 54 + !6 ,7 4 - 8

3 1 4. + 6 , !" 19 - !" 9" . " : : ;; 1 0 <" . 0 " + 1 2 =8 ;> : + ? 3 + 4 ! x@ • x . 7 5 ? x ≡ 5 mod 7 • 8 2+ 3 + + ? 6 x ≡ 4 mod 5 3 x = 4 + a · 5 = 5 + b · 7 + a b a · 5 = 1 + b · 7 + 7" a · 5 = 1" a ≡ 3 mod 7 ?" s" a = 3 + s · 7 A 6 x = 4 + (3 + s · 7) · 5 = 19 + 35 · s

x ≤ 35 x = 19

a

! b

! " # $ %

& ' (

% %

35 45 ) * + ,-

./ + ./ "0 1 2 x ≡ 2 mod 7 x ≡ 5 mod 93 2 x ≡ −1 mod 3 x ≡ 3 mod 43 2 x ≡ 2 mod 6 x ≡ 5 mod 93 2 x ≡ −1 mod 12 x ≡ 1 mod 14

,

,9

,=

4 % 5 0 999 6 8 $ a 125 b (

% 7 5 a b % 8 a = 7 b = 5

6 " 5 : ; : 7 " ( 88 # 6 "

< 2

( $

( # 88 % $

( (

5$ &88 1, 2, 3 4 6 ( < >? 1 % 7 %7 / @2

2 ./ $ % a1 x ≡ 1 mod 2 x ≡ 2 mod 3 x ≡ 3 mod 4 x ≡ a mod 5 (

a ∈ {0, . . . , 4} % 8 ./

! " 1) x ≡ a mod m 2) x ≡ b mod n (a, b, m, n ∈ Z, m, n = 0) #

• $ %& ' • $ %& ( '

•

x = a + r · m = b + s · n b − a = m · r + n · (−s) r, s! "# $ % ! ggT(m, n) & '" b − a ( r, s ) ) * " +( , "

- m n '" ( . + / ) * " ! 0 1 r, s! "

m · r + n · s = 1, m · r · (a − b) + n · s · (a − b) = a − b, x := a + m · r · (b − a) = b + n · s(a − b). . x = a mod m x = b mod n( . 2! x ( ) " 3 4

m, n a, b x ∈ N x < m · n x ≡ a mod m x ≡ b mod n

x

x ≡ 1 mod m x ≡ 0 mod n

x x ! m · n" # " 0 ≤ x < mn x x $ 1

x1 = a + r1 m = b + s1 n x = a + rm = b + sn

(r − r)m = (s − s)n = x − x $ x − x m n " mn % (x − x) x = x . & # '" ( & " ) 2 1

1

1

1

1

1

a, b, m, n ggT(m, n) = 1 x

(a, b) 0 ≤ a < m 0 ≤ b < n x x ≡ a mod m x ≡ b mod n. ! " #

Z/mZ × Z/nZ $%

f : Z/mZ × Z/nZ → Z/mnZ f (a, b) ≡= a mod m f (a, b) ≡ b mod n $%

f (a, b) : = a + m · r · (b − a) = b + n · s(a − b).

&'()

! r s # 1 = mr + ns * mr = 1 mod n ns = 1 mod m " + , $% $% m = 5 n = 7 - ' .

&1) & 1) &'1) &.1) &1)

&1 & 1 &'1 &.1 &1

' .

' 2 '(

) ) ) ) )

' &1') & 1') &'1') &.1') &1')

/ '' ( '3

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. &1.) & 1.) &'1.) &.1.) &1.) . . 2 . '

&1) & 1) &'1) &.1) &1) '/ .' (

/ / '0 ' .. 3

/ &1/) & 1/) &'1/) &.1/) &1/)

0 &10) & 10) &'10) &.10) &10)

0 ' 0 '2 . .

! $% chines(a, b, n, m) - *#4

$ - 5 Z/mnZ

% 6 6 6# 7 - 5 8 3 - 7 8 4 # +

chines : Z/mZ×Z/nZ → Z/mnZ chines Z/mnZ chines n m ! " #! f (a, b) chines(a, b, m, n). $ a, a ∈ Z/mZ = {0, 1, 2, . . . , (m−1)} b, b ∈ Z/nZ f (a, b) = f (a , b ) " b+n·s·(a−b) = b +n·r·(a −b ) b ≡ b mod m % a ≡ a mod n $ x ∈ Z/mnZ a :≡ x mod m b :≡ x mod n x = f (a, b) " ! " # $% & ' ( $ r, s n · r ≡ 1 mod m m · s ≡ 1 mod n x ≡ a · n · r + b · m · s. )

) chines & *

+ ,!(

+ x ≡ 20 mod 35 x ≡ 28 mod 36+ x ≡ 10 mod 19 x ≡ −2 mod 28 + x ≡ 4421 mod 5891 x ≡ 11800 mod 16200+ 3x ≡ 5 mod 77 x ≡ −6 mod 12 + 5x ≡ −3 mod 11 −3x ≡ 5 mod 13+ x ≡ a mod m x ≡ b mod (m + 1)- ' .

/ 0 1 !2 3 99999 0 1

0 9 3 0 1 49375 0 5 4

5

+ ,! 6 ' [−1000, +1000]( x = 2 mod 12 x = −1 mod 21 + ,! 6 ' [−200000, 200000]( x = 51 mod 255 x = 120 mod 247

+ ,! 6 ' [−900, 900]( 3x = 2 mod 5 11x = −3 mod 14

x = a mod n x = b mod m [c, d] ! " # $ % & $ ' ( ) * " m((n(+ $ m, n , - m, n > 1) $ ( " . 53747712 = 6561 · 8192 = 38 · 213 " , / 0 + 6561 1. 8192 1.

/ 2 $ + a 2 $ + b 3 2 4 " 1 53747712( + 17432577 · b − 17432576 · a 2 5 ,6 " .$ $ 6 "

# .. # 5 / 7 . ( # * 8 5 " 5 &$ $5 m n $ 5 6 $ m n $ '5 ' 7 + ) x ≡ a mod m x ≡ b mod n x + kgV (m, n) · k " 9 x ≡ a mod m x ≡ b mod n : x ≡ 17 mod 40 x ≡ 7 mod 25 ,2 1 ;< $ 1 <=) x ≡ 17 + 40 · k ≡ 7 + 25 · l 40 · k − 25 · l ≡ −10 2 19 6 $ ! ' 19 x ≡ a mod m x ≡ b mod n , % $ * m · n 19 , % $ '5 ) ' " 18 24(+ $5 / 2 5 " $ m · n(+ ;<

/ /- m = 21 ( n = 52 /- $ * ! 3 / # ' # / + 5 $ % * /- 0 20 5 0 51 8 > / ' ? / 5 4 0 $ * ! 5 +- / 4 17 , +

11 ! " ! # 54 "$ # % ! 21& ' $ ! ! ( ) * +,-& . # # # % # # # $ & ' #

52 : 21 54 : 21 # # " *

" # +-+&

/ 0 1 2 $ ) # . 3 88, 225 365 4& # .

# # # # # 15, 43 !& 100 4&

2 $ ## . % 5 # $ ## # 2 ) #$ 2 . & +,-$ . ) # & +,-

! 3 # 6 ! 4 ! # !# # ! #" ! # # . # # $ ## # 1 3 # 6 +-7& 0 ! "#5 # 89 :

/0 ; /0 < ==/0$

.>3 ; .>3 < ===.>3& ? # # *! # #% # # ! & +-@& 0 # * 5 /# * A x2 = x mod m. . x # .# !

9# . m (0 < x < m)& m = 2, . . . , 50 9 . m& ' 9 . # !# 2 # 1 $ # m * 9 .% m #& ' ) # * " # ' ! # ) # m !% #$ ## ! # 1 # & ' ! ! # !# # m & # x2 = x mod 11 0 : x·(x− 1) = 0 mod 11 $ ## x2 = x mod p ! #

p$ ! p 1 #& # x2 = x mod 81 * ## 1 % 9 &

21 x2 = x mod 21 x2 = x mod 21 x·(x−1) = 0 mod 21 x · (x − 1) = 0 mod 3 x · (x − 1) = 0 mod 7 (x = 0 mod 3 x = 0 mod 7 (x = 0 mod 3 x = 1 mod 7) (x = 1 mod 3 x = 0 mod 7) (x = 1 mod 3 x = 1 mod 7). 1) x2 = x mod 77! 2) x2 = x mod 77! " x2 = x mod 675 # p, q $ % & pr · q s $ $ ' ( )*+ , - $ )*" ./'0 5678·5678=444445678 123·123=4444123 )*9 x2 = 1 mod 10! x2 = 1 mod 100! x2 = 1 mod 1000 )*: ; # ' & < # 1

= # $ > 5 ? , @( 0 = ; $ A & & B A & $ & & $ ; C # 1

> 100 $ < 8 7 ' 9DE 0 ::9

? ; &

1 6, 5, 4, 3 5, 4, 3, 2 6 ; ' 7 F6 )GGG 5

<

7 ; & 1 2, 3, 4, 5, 6 1 A 7 F6 3 = ( 0

.

$ . 7 .

8 B < H =

; # 6 IB 0 . $ A ( & # 1 + 24 + . . . + n4 , 0 #& # $ 'A = A < ; 7 'A $ % @

C

(

$ = )+":F)+*:

7

0 ; A 1 10, 13, 17 3, 11, 15J

! "#$ $ % # & '$## $ ( ) *

+ ,- # # % , . # Z /. Z[φ] 0 "#$ 1 $ 2 / 3

4 5 6 7 % # # $

3 , # - . "#$ % ($

# - ,- 8 a (= mn , m > 1, n > 1) 2a 8 3 ) # $ a = 4 2a = 8 9. a = 3n 2n > 1 # %

%.5 3a 8 3 $ 8# 8 a 2 · a 3 · a - 8 2" ,- # 5 % # $

- , ) -( a = 2r · 3s , 2a = 2r+1 · 3s , 3a = 2r · 3s+1 . 3# a, 2a 3a 8 #( ggT(r, s) = d > 1 ggT(r+1, s) = e > 1 ggT(r, s + 1) = f > 1 3

a, 2a, 3a 8 # :. d > 1, e > 1, f > 1 8 -( - r = 14 s = 6 2 r = 6 s = 145 $ #) 8 3

a = 214 · 36 = 11943936 = 34562 $ 8 2a 3a 8 2 2a = 2883 3a = 127 5 % I " ( 9 1 ;# ( 9 a i · a -( i ∈ I 8 < .- -( I = {1, 3, 4, 5} -( 3 +-

.# # %

$ #) a - # = $ - I >? a = 3r · 4s · 5t r, s, t

(∗)

ggT(r, s, t) > 1

ggT(r + 1, s, t) > 1 ggT(r, s + 1, t) > 1 ggT(r, s, t + 1) > 1 8$ P 8# P = 2·3·5·7 = 210 3

- r ≡ 0 mod P2 @

s ≡ 0 mod P3 t ≡ 0 mod P4 r ≡ −1 mod 2 s ≡ −1 mod 3 t ≡ −1 mod 5 7, 2, 3 5 > 1 ! ! "# r = 105 s = 140 t = 84 a = 3105 · 4140 · 584 a1 = 3105 · 435 · 584 $ % & a1 ' (

$ ) * I ⊂ N +& , a ∈ N i · a - i ∈ I . ! * . ! a - n ∈ N +& 2 · a, 3 · a, . . . , n · a . ! ( )

a 2 · a, 5 · a, 7 · a

a 2 · a, 3 · a, 4 · a, 5 · a I ! "

#

" $ % & ' n n(

) M ' & * ! ! ' ∈ M $ a a 2 · a 3 · a 4 · a, . . . , 12 n · (n + 1) · a + * ! ! M = {a, 2 · a, 3 · a, . . . , n · a} ! % i · a

i ∈ {1, 2, . . . , 12 n · (n + 1)}. " M ) % , ) ' x, y, z x y z x + y x + z y + z x + y + z -. /) / ( 0 ! 1 2 3 $ 4 % - & n n/

) * ! & ' ) 5 - $ k ( (

k > 1 ) {6, 19, 30} {407, 3314, 4082, 5522} {7442 28658 148583 177458 763442} (k = 2) {63, 280, 449} (k = 3) ) 6

7

)8 -9 : ! %6 k = 2 * ! n = 6, k = 2 ! *; n > 6 < = + : .* ;

; > ; ? 0 " " @: - ,# , (,A

/& $ ! +& +& ! ! -+/ % 0 +& 1 +& +& 2 !3 & !$ 2 ! 4 5 ( 67 8, 9 m & !$ +& . & 4 1 & $ #0 5 "# +& / $0 : & 2 0 "# 0)

0 1000 7 a 11 b 13 c a, b, c ! ! ! "# $%

(∗) x ≡ a mod 7;

x ≡ b mod 11;

x ≡ c mod 13.

x [0, 1000]. & 7·11·13 = 1001 x ∈ [0, 1000] '7, 11, 13 ( !) * + , $ # - ' .

! , /!

1001 ∈ [0, 1000] )% & % $ ! - - ! 0+ ! * !% 1 = 2 · 11 − 3 · 7 x x = 22a − 21b + 77k ! $ . x = c + 13 · l 0 ! 22a−21b+77k = c+13l 13 k = −c+9a−8b 1

x = 22a − 21b + 77(−c + 9a − 8b) $ ! 1001 & 2- % 3 "

m1 , . . . , mn a1 , . . . , an x = a1 mod m1

=

x = an mod mn

m1 · . . . · mn 1 * /! 2-

! 0 ≤ x < m1 · . . . · mn - 2- ! n = 2% & x ≤ x 2- ! 1 mi , (i ∈ {1, . . . , n}) 4 x −x 5 mi 4 x −x 0 ≤ (x −x) < m1 ·. . .·mn ! x −x = 0

m1 · . . . · mn - 2- &

M 2- & % M = m1 · · · mn Mi = mi

Mi mi i bi bi · Mi = 1 mod mi

x = a1 · b1 · M1 + . . . + ai · bi · Mi + . . . + an · bn · Mn ! " # mi Mj = 0 j = i x = ai · bi · Mi = ai $ % bi bi · Mi = 1 mod mi 2 & ' ' ()) * ' % * + , % , "

a = 5, b = 6, c = 8 (∗) ! x ≡ 1 mod 5" x ≡ 3 mod 7" x ≡ 5 mod 12 ! x ≡ 109 mod 210" x ≡ 4 mod 1155" x ≡ 389 mod 5005

# $ ! % & " '( " ( ) x ≡ a mod 2" x ≡ b mod 3" x ≡ c mod 5 * a ≡ 0" b ≡ 1" c ≡ 3+ x ≡ a mod 3" x ≡ b mod 5" x ≡ c mod 7 * a ≡ 1" b ≡ 4" c ≡ 2+ x ≡ a mod 7" x ≡ b mod 8" x ≡ c mod 9 * a ≡ −2" b ≡ 1" c ≡ 3+ x ≡ 5 mod 16" x ≡ −4 mod 9" x ≡ 9 mod 13+ x ≡ a mod 3" x ≡ b mod 5" x ≡ c mod 7" x ≡ d mod 11 *'( ! a ≡ 1" b ≡ 2" c ≡ 5" d ≡ 7+

x ≡ 2 mod 8" x ≡ 3 mod 81" x ≡ 4 mod 25" x ≡ 5 mod 11 , '( - x ≡ a mod 7" x ≡ b mod 11" x ≡ c mod 13 % ( ( # $ ) . ( / 0 ( ( # $ ( *% $ ( % ! % $ " ggT(M1 , . . . , Mn ) = 1 1 $ '( k1 , . . . , kn " 1 = n

n

i=1

ai ki Mi

i=1

2 ! x ≡ 1 mod 2" 2x ≡ 1 mod 3" 3x ≡ 1 mod 5+

ki Mi ' ! x =

x ≡ a mod 2 2x ≡ b mod 3 3x ≡ c mod 5 2x + 1 ≡ 0 mod 3 3x − 2 ≡ 0 mod 4 4x + 2 ≡ 0 mod 5 x − a ≡ 0 mod 3 3x + b ≡ 0 mod 5 2x + c ≡ 0 mod 7 a ≡ 2 b ≡ −c ≡ 1 3(x − 2) − 1 ≡ 0 mod 4 2(x − 3) − 1 ≡ 0 mod 3 2(x − 4) − 3 ≡ 0 mod 5

ggT(a, n) = 1

x, y ∈ Z 1 = ax + ny n

a · x = 1 mod n x a · x ! n 1 " a # Z/nZ # n $ ≤ n % n ∈ N ϕ(n) &

n $' ≤ n &

# ( Z/nZ ( )

n 1 2 3 4 5 6 7 8 9 10 .. n ϕ(n) 1 1 2 2 4 2 6 4 6 4 .. ? * ) )+

n ,$ ) " + n - * . n = 24 Cd = {x | x ≤ n ggT(x, n) = d} / n = 24 )+

Cd

# ggT(x, n) = d $ x ≤ n

ggT( xd , nd ) = 1 nd ϕ( nd ) $)

C1 = {1, 5, 7, 11, 13, 17, 19, 23} C2 = {2, 10, 14, 22} C3 = {3, 9, 15, 21} C4 = {4, 20} C6 = {6, 18} C8 = {8, 16} C12 = {12} C24 = {24} * 0 $ " &

Cd 24 1" #

ϕ(n)

24 ϕ(24) ! " # ggT $ % x ≤ n n & !

n n = ϕ(d). n d|n

ϕ(d) d n Cd = {x | x ≤ n ggT(x, n) = d} # {1, 2, . . . , n} =

Cd

d|n

' (

) Cd d n * & + 'Cd ) = ϕ( nd ) # ) Cd , 2 . ϕ(m) m /+ , +

p n ∈ N ! ϕ(pn ) = pn · (1 − 1/p) = pn−1 · (p − 1). 0

pn = ϕ(1) + ϕ(p) + . . . ϕ(pn ) p

n−1

= ϕ(1) + ϕ(p) + . . . ϕ(pn−1 ).

& ! ϕ(pn ) = pn − pn−1

2

m n + ϕ(m · n) 1 + 2 + 3

chines(a, b) := a + m · r · (b − a) = b + n · s · (a − b) # m · r + n · s = 1. # & 0

m n ! ggT(a, m) = 1 = ggT(b, n) chines(a, b)

m · n

" + ggT(a, m) = 1 = ggT(b, n) &4 c = ggT(chines(a, b), m · n) # chines(a, b) = c · d m · n = c · e d, e ∈ N 0

chines(a, b) · e = c · d · e = c · e · d = m · n · e chines(a, b) · e ˙ − b) · e = m · n · e = a · e + m · r · (b − a) · e = b · e + n · s(a a · e ≡ 0 mod m b · e ≡ 0 mod n. ggT(a, m) = 1 ggT(b, n) = 1 a Z/mZ b Z/nZ e = 0 mod m e = 0 mod n m n e m · n e k e = k · mn mn = c · k · mn 1 = ck c = 1 c ggT(chines(a, b), mn) ! ggT(chines(a, b), mn) = 1. x ∈ N

chines(a, b) · x = a · x + m · r · (b − a) · x = b · x + n · s(a − b) · x = 1 mod mn. 2

ax = 1 mod m bx = 1 mod n

m, n ϕ(m · n) = ϕ(m) · ϕ(n).

" # $ % chines & ! ' ( (a, b) ∈ Z/mZ × Z/nZ ggT(a, n) = 1 ggT(b, m) = 1 m · n % )

Z/mnZ mn % ) (

ϕ(m) · ϕ(n) 2

n = pr11 · . . . · prkk ϕ(n) = p1r1 −1 · (p1 − 1) · . . . · pkrk −1 · (pk − 1) 2

" * % n %

Z/21Z Z/63Z Z/49Z 7 · x + 14 = 3 Z/45Z ϕ(10n )

n ϕ(n) !

" # $ % n & ' < n

n 2

5 3327 ϕ(n) ! " " # !$ ! % ! ! & p n

! ' ! p ( pk

n "

! ϕ(n) = (p − 1) · pk−1 · ϕ( pnk ) ! )! ϕ(n) ! ϕ(n) * ++ ! # ! !! % ! " & ! ( ,

-%

. /% ! ! 0 ! 12 "* ϕ(n) = ϕ(n + 1) 3! ! &

! ! )42 5! ! 2 ! 06 -* ( '! 3 & - ! * . 2 ! 6 3 ( ! +7 % ! " # $ n d " 8 ! - a, a+d, . . . , a+ (n − 1) · d ( 9! - n

9 ' .! α1 , α2 , . . . , αϕ(n)

* < n " n !

n " " α1 α2 . . . αϕ(n) n + α1 n + α2 . . . n + αϕ(n) 2n + α1 2n + α2 . . . n + αϕ(n) ... ... ... ... (m − α1 )n + α1 (m − α1 )n + α2 . . . (m − α1 )n + αϕ(n) : $ 8 )" n m ! nm " -* ( ! !

; ( ! +< ( !" (! " ! !

! ( !( (

( (! = > ( ! 3 =?2 !!> @! ! ! ( A 3 . & 2 !! ! B . R . S ϕ : R → S ϕ(x + y) = ϕ(x) + ϕ(y) ϕ(x · y) = ϕ(x) · ϕ(y) "*

x, y ∈ R @ % " Z/mZ × Z/nZ = {(a, b)|a ∈ Z/mZ, b ∈ Z/nZ} 2 ( ! B C 2 $ (a, b) + (c, d) := ((a + c) mod m, (b + d) mod n) (a, b) · (c, d) := ((a · c) mod m, (b · d) mod n)

R := Z/mZ × Z/nZ ! " # R$ "

f : Z/mnZ ∈ x → (x mod m, x mod n) ∈ R % chines ◦f = IdR f "

% & e Z/mZ " & f (e) R "

' () * +" * "

,

' * # , ! - & . m · n " m, n * , 0, 1, 4 . 5 0, 1, 2, 4 . 7 ! . 35$ " , m n & m n & " m · n

/)

, 0- 60 * " (1 x2 = x mod 602 x2 = 1 mod 60 3 * " " ! 700 x2 − x = 0 mod 7002

x2 − 1 = 0 700

, 210 4 5! 6 1

# 78 9 2 " - : ; " 9 *< = 78 m$ /> , ? * " (1 4 m = pr11 · . . . · prnn 7 ; * m& " 2n 0- m

! "# $ % & ' & () * ) + n n

) + , ' -& & ". /& ,

0) -& & , -& , 1

+ 2

3-& & ) 4 05

3#

n

n ! " # ax + b (a = 0, x = 0, 1, 2, . . .) $

% 2n

! p1,1 . . . p2,n a # & ' ! a ! # & ( )

a · x + b ≡ 0 mod p1,1 · p2,1 a · (x + 1) + b ≡ 0 mod p1,2 · p2,2 ... ≡ ... a · (x + n − 1) + b ≡ 0 mod p1,n · p2,n

x ≡ −a1 · b mod p1,1 · p1,2 x ≡ −a2 b − 1 mod p2,1 · p2,2 ... ≡ ... x ≡ −an b − (n − 1) mod p1,n · p2,n

%* # i ∈ {1, . . . , n} ai · a ≡ 1 mod p1,i · p2,i +$ ggT(a, p1,i · p2,i ) = 1., & - " .)

/ 0' x 1 p1,1 · . . . · p2,n " % # &

2

n x, . . . , x + (n − 1) !

! # ( n ! " # & # 2

! " ! " ! # $ % &

" ! $ ' " ! " % ! ! (

) ' * # $ % &

" ! + ,!!- . / x = ((n + 1)!)2 + 1. $ x + 1, . . . , x + n % &

" ! 01 " ! ! * * 2 3

x, x + 1, x + 2, . . . , x + (n − 1) x + 1, x + 2, . . . , x + n x = (n + 1)! + 1 ! " # " " $ $ %& '

( )* + %$ , +- "

# .$ $ /$ $ 0 ( # 1 2 ax + b ggT(a, b) = 1 # $ 1 0$ $ . ' # ( 3 )* 4 5 "0 6 $ 7- # ( 89: 1 " 4" $ "0 " , ; 4 . < = & 0$ * 0 > ? " # @ > 1 =$

30 ? "

$ * ( " =$ "0 2 0 n n " " 0 ? " A 0 2 n $ n " " 0 ? "

=$ "0 n = 3 n = 4 n = 5 "0 $ $ 1 < 0 ? " π62 0 x > 25 $ 0, 1 · x ? " $ x " ( k n 0 $ n " " 0 k > 1 $ 7 $ " " $ $ 7 %

"0 4"0 $ 2 n n = =$ 2 n

n = $ " k > 1A

( " "0 k = 3 n = 4 k = 10 n = 2 * $ 899 d # # # 2, 5, 13 = d > 1 "0 2 # a, b B {2, 5, 13, d} 15 a · b − 1 ? " < $ # dA

½¾¼

{2, 5, 13, d}

a, b a · b − 1 ! "#$% &''

( ) * + ,

- + + . - ( * ) ) + / 0 ( * 1 + 2 3 a < b ) 4 ) 5 a + n b + n + , a < b < c < d ) 4

) a + n, b + n, c + n, d + n 6 + , 7 n 2 + n, 4 + n, 24 + n 6 + 8 a < b < c ) ) n a + n, b + n, c + n 6 5 + ggT(a + n, b + n) = ggT(b − a, b + n) ggT(a + n, c + n), ggT(b + n, c + n) p1 , . . . , pr 5 1 + b − a q1 , . . . , qs 1 + c − a r1 , . . . , rt 1 + c − b 9 + . : 5 ;+ b + n = 1 mod p . . . < i + j + k : ) <+ ggT(b − a, c − a) > 1 q = p b = c mod p 9

+

) =) : + + . >

&'" 4 6 1 + : 2 5 4 6 8 8+ Z2 8 1 P > + 4 6 Q +

? [P Q] > P Q 4 6 +

A(0, 0) B(1, 0) C(0, 1) D(1, 1) 4 6 3 + A B C D 4 6 E 4 6 , 4 6 Q(q, kq + 1) ) k, q ∈ Z + 2 6 (0, 0) ? A B C 4 6 4 6 D 1 A B C @ A 5 + x5: 1 A B C y 5:

+

3

"B# > "$' + + .@

Zn ! "# $ " % & '( #

Z/12Z 0 12 2 0 → 2 → 4 → 6 → 8 → 10 → 0 {0, 2, . . . , 10} = 2 · Z Z/12Z U !" # u ∈ U U

uZ = U

0 < m ∈ N a ∈ Z/mZ Z/mZ a m

a, m $ % x, y ∈ Z 1 = ax + my & b = axb + myb ' ( b ∈ Z/mZ % b = abx ) " a !" # Z/mZ

% x ∈ Z 1 = a · x mod m & % y 1 = a · x + m · y a m $ 2 )$ " ϕ(m) % & ) Z/mZ

&$%

))# * !+ $ ,( Z[φ]# %- #

! " " #

# $ " " #% &

" " # ' #

(

) * + *, " """" -% #% ./0 123456 2758

9

+ " " 9 $ " : " . ; -

36 ! " #$% % #$% & ' (%' ) $ ! % ' ! % # % ) ! ' * 19 + * )' + , ' $ - %

2

=

7

...

25

23

=>

=2

==

...

74

+"" "

=

?

4

...

74

2

7

@

A

...

7@

+""

"

<) <)

./ % & ! ' ! % + , / & 0 0 ' !' a (a ∈ {1, 2, . . . , 36}) 1 / 2 ! 34' 5 6 a∗ ' / a∗ = 2 · a mod

37 (1 ≤ a ≤ 36). n a

a∗ (1 ≤ a ≤ 36) a∗ = 2n · a mod 37. n a∗ = a a a = a∗ ≡ 2n · a mod 37. 2n = 1 mod 37. ! " n (n > 0) 2n # 37 $ 1 %& '

(%' 36 ) $ 37 36 " 2 · 1, 2 · 2, . . . , 2 · 36 & * + + $ " # 1 36 , " $ & - .' $ # 2 · 1, 2 · 2, . . . , 2 · 36

" # 1 36

& / (2 · 1) · (2 · 2) · (2 · 3) · . . . · (2 · 36) = 1 · 2 · 3 · . . . · 36 mod 37. % / 0 1 2 1 % n! ≡ 1 · 2 · . . . · n)

'

236 · 36! ≡ 36! mod 37. (' 37 " # 1 36 36! 37 236 = 1 mod 37. " 2 3 . 36 / 0 % . $ & + %

/ 0 &

% $ # 2n 37 n ≡ 1 36 4 . . .4 ' .% 5! - 236 ≡ 1 mod 37 % # 6

0 = a ∈ Z/pZ

p

ap−1 = 1 mod p

a

ap − a

p

a

Z/pZ \ {0} x → a · x ∈ Z/pZ \ {0}

(1a)(2a)(3a) · . . . · (p − 1)a = 1 · 2 · 3 · . . . · (p − 1) mod p (p − 1)! · ap − 1 = (p − 1)! mod p

p (p − 1)!

ap−1 = 1 mod p. a ap ≡ a mod p p a ap ≡ 0 ≡ a mod p. 2 ! " #

$ " % &

p 2 · p ap − a m5 m m ∈ N ! "

# ! $ % $ & ' ( ) * & + , 652 = 650 · 62 = (610 )5 · 62 = 36 = 3 mod 11 - ( ! , 20350 : 7 382 : 17 6100003 : 101 217 : 19 . p−2

2 : p p = 2 (270 + 370 ) : 13 / &

% - , % m ∈ N , 42 m7 − m 1 1 7 · m 0 · m5 + m3 + 5 3 15 1

- 2 3 4 5 &

3+ 6 6 4 -

3 7 6 4 -

3 4 7 8 4 -

3 - ! 3 + 9 3 ! :; < $ $

3 0 8 3 - + $ = 3 - + ! ; $ > < < 3 & $ $ >

? - + 2 -+ $ 4 - + 4 5 8 4 ( 3

3 - + !

! "#$ % & ' ()*

t, ! t t+ + t t! + t = = t t t

!

,

,

! - a % + a = 2. & - p - p - + / - 0' 0' 1 - 2 3 0' 2 1 - ' 0'. & 2 & 2 2

& ' 0' 1 - p 4 ' %5 6& %0. 4 7' ' 0 ! / p - & ap − a +a 2 0' - p / p 8 2 ap − a ap = a mod p 4

&

' 3 9' : + 3' ; %"#<=& 1 > ; ' 1 ? 1 5@ A . & / p ' ' p - ! 2 2 ! ' p ' 6 %B & ' 6 C 6 2 4 4 ' ' & "#= % & "#= %& p ' -

! " # $ % & '( m n) ( n * m) # n! n n · (n − 1) · . . . · (n − m + 1) = := 1 ·2 · ...· m m! · (n − m)! m # + (a + b)n = an +

n

n n−i i n n−1 n n−2 2 a b. a b+ a b + . . . + bn = i 1 2 i=0

' , ( $ )# # * + p - * m, 1 ≤ m < p :

p = m

0 mod p.

# . (a+b)p = ap +bp mod p 'p -# / Z/pZ (a+ b)p = ap + bp '+ Z/2Z . (a + b)2 = a2 + b2 0 , * 1# 2 3 4 p 4 5 - p 6 2 4 (. 7 86 ) #

* n 9 (a1 + a2 + . . . an )p = ap1 + ap2 + . . . + apn mod p

# . * a1 = a2 = . . . = an = 1 : np = n mod p. :

# ! (9 4 % ) + p - (p − 1)! = −1 mod p '8 2 Z/pZ -; X p−1 − 1 8 . < # # + = 0

:: p - q = (p − 1) · t + 1 - t ∈ N t > 1 ' q > p# % 2p·q = 2p mod pq 0 : > 6 '( 6 )# 2 / > 6 2 n * Rn 10n − 1 Rn = 1111 . . . 111 'n # = " . 9 * > 6 ' 4 2 #

Rn ! " # $ % & &' 4 ( ) * ! + % , - x 7 · x . / 0 x * - 13 1 0 p > 5 Rn 2 ' p p Rn 3 3 4 % ) % 5+ 15873 6 15873 5 6 ) % 5& 4 8 7 3 56 8 - 6 / 7 . / . * " ! 8 9 12345679 3 2 ' 9 (9, 18, . . . , 81) 9 : $ 7 ; * + ' / -

p ! "

"# Z/nZ # # !

n>1∈N

a

n

aϕ(n) = 1 mod n

$ a n {a · x | x ∈ Z/nZ} %

! &' a·x # Z/nZ ( # Z/nZ ! )*# # + !, &

+ # a · x! a1 . . . aϕ(n)

+ # Z/nZ a · a1 , . . . a · aϕ(n)

+ # !

a1 · . . . · aϕ(n) = a · a1 · . . . · a · aϕ(n) a1 · . . . · aϕ(n) = aϕ(n) · a1 · . . . · aϕ(n) 1 = aϕ(n) mod n &# <=> 7 2 )

!

" # $ %

& n < 36 ! 2n = 1 mod 37 ' ( )

! *

+# ) ,- # !

. / , e & 1 36 0 / e, 1 ≤ e < 361 2e = 1 mod 37 ' $ 36 e/ 36 = k · e + r, 0 ≤ r < e / 1 = 236 = 2k · e + r = (2e )k · 2r = 2r mod 37! 2r = 1 mod 37 ' 0 ≤ r < e! e & , 2e = 1 mod 37 ! r = 0 ' ! e + $ 36 ' ,

p

a p n an = 1 mod p e ae = e n e p − 1

1 mod p

a = 2 p = 37 0. 21

2

3 4 5 + $ 36 + $ 12 $ 18 , ! " $ 212 218 / 212 = 26 mod 37, 218 = 26 · 26 = (−11) · (−10) = −1 mod 37 ' n < 36 : 2n = 1 mod 37 ' 6 77 3 & e > 0 ae = 1 mod p 8 $ a p / p (a) ! 8 & p + $ p − 1 ! 37 (2) = 36 = 37 − 1 ' 6 79 3 ( p (a) = p − 1! a ( $ p , : / % ; ( p ( $ < & : ' %= /

p > 2 m2 + 1 4 p − 1 ggT(a, b) = 1 a2 + b2 = 0 mod p p = 2 p = 1 mod 4 ! " # 3 36 37 $ %& p = 2, 3, 5, 7, 11, 13 17 1 < a < p−1 " a p ' " # a p ( ) a a2 a3 . . . , ae * ' p e " a p + , ) " e a p # ae/2 p - a p Z/pZ \ {0} = {an | n ∈ N} .

/ 0 Z/pZ 1' 2** 3 a 4 5 6 ' 3 5 7 11 . . . 31 7%' 7 p ' " (p−1)

a p a q = 1 mod p & q p−1 & & a p−1 p a 2 = −1 mod p p ' p 8 9

) ' : & ' p(p > 2) p + %& p p − 1(= −1) p+

d Z/pZ p ! " d # $ p − 1 Z/pZ # d % & !

2 ' ( Z/7Z) * 6 ( # + ! " p ≥ 3

, 2 +- . X 2 − 1 = 0 = (X − 1) · (X + 1) ! % /0 $

0 1 −1 = p−1 mod p 1 1 ! " #$ $

%# &

Z/pZ

p d p − 1 d d Z/pZ

' y d y ( X d − 1 = 0 ! ( )% d 2 * % $

p 0 ≤ d ≤ p − 1 Z/pZ ϕ(d) d p + Z/pZ d $

$ a ∈ Z/pZ d ' M = {1, a, . . . , ad−1 } , " % - " xd − 1 ! " )% d " xd − 1 M ! % . d (% * ,, M / 01 (% * ,, d ϕ(d) ! $ 2 , 2 ! %

p d p − 1 d Z/pZ

ϕ(d)

Z/pZ \ {0} = {1, . . . , (p − 1)} % Ad := {x | p (x) = d} ! |Ad | := (Ad ) = 0

|Ad | = ϕ(d) 3 {1, . . . , p − 1} = Ad ! $ (p − 1) = ϕ(d) =

d|(p−1)

d|(p−1)

|Ad | / 0 # |Ad | = 0 # % /

d|(p−1)

|Ad | = ϕ(d) $& 4 d " p − 1

p

2

Z/pZ ϕ(p−1)

! " # $ % & & ' $& (& % ) % & ' * +& % ) , &% -.- !

/0 / .1/ 2& % , ' & /0 3

4 , % % " 5 = 1% 6 % (& &' (& p & %

(& 7 2 8 (& p = 3% 5% 11% 13% 19% 37% 53% 59% 61% 67% 83

+ & 84 9923 9941 9949 $& &% 2 (& && 9&& & 2 '% & 5 2, 3 5 (& (& p

& $& & & :; <" (& => , =& , ? +@ , & 9 % $ 0% A B + .--4% 88 C. ' 1D4 * 1 9949 1

9949 !

" #" $ %

& ' ( ) " !* ) 5, 7 17

"

n ! !*

+ , - ." ! 2 ! * / 5, 7, 17, 23, 29, 31. 0 1 2" ! 1000

" 2 !* 3

!* / 3, 5, . . ." 101 $ 10 !* & 4 ! 5 p = 4 · t + 1 / a !* " −a !* 67 1 p ! 5 4·t+3 / a !* p " p (−a) = p−1 2 68 ( ! 2, 3, 6 !* & $ " 9

15 + 25 + 35 + . . . + 65 7 15 + 25 + . . . + 105 11 15 + 25 + . . . + 165 mod 17

! "

! #! $ % p−1

p > 3

i3 = 0 mod p i=1

p−1

i4 mod p

i=1

& #! ' ( )

p−1

ik mod p

i=1

"* + #! , + - (p − 1)! = −1 mod p " . + /

+ - 01 20 " 3 (4 + 5 # + !$ ( # 6 78 #! 09 " : ( ; & . 3 (

4 <" " $= 1

5' > x7 = 1 mod 29 ?

5' > 1 + x+ x2 + x3 + x4 + x5 + x6 = 0 mod 29 9 5' ! >

1 + x2 = 0 mod 49

1 + x4 = 0 mod 49

1 + x8 = 0 mod 49

! " #

$ % &

' ( % % % ) $ % & % * +* , 1 , - & $ = 3

1 0, 3333 . . . = 0, 3 1 = 0, 142857 6 7 2 5

a

!" p p = 5 #p a$ #!

% 1p $ & ' (" % !" 1

) * p 1 z = = 0, a1 . . . al = 0, A A )+ a1 . . . al l p & A

a1 · 10l−1 + . . . + al 10l − 1 10l ·z−z = A z·(10l −1) = " ) 10l = 1 mod p p l ) ( l , % - . % p − 1 & ( /

l %

0 l . % p − 1

1 , % 10 p p 2

' ( /

l = p (10) l p

1 p − 1 10 p

!" l = p − 1 l < p − 1 # $ !" %& & p $ p − 1 ' %& $ %& %& ! %& p − 1 % ( ) ! * %& + + "

1 %&%$ p , & p %&%$ - . 4

- . 10

1 p

-% . 7 &

%& %&%$ ' ( ./ 0 ! *! $ ( ' a ≥ 2 ' n ∈ N & p (a) = n 1

p p (2) = 4(5, 6, 7, 8, 9, 10) n ∈ N p (2) = n 2n 22 + 1 ! 2n · k + 1 " n ∈ N p ! p1 # 2n $ a ≥ 3 " n ∈ N p (a) = 2n p (3) = 3 %9, 27, . . . , 3n p (10) = 3 %9, 27, . . . , 3n & ' ' (

# f (X) = X 2 + X + 1 ' n n ∈ N ggT(f (X), f (X 3 ))

$ x ≥ 2 n ∈ N p p (x) = 3n ) ) *+, % 5 7 " - ' # ./,* $ a ≥ 2 n 0 p (a) = n ) a = 2 n = 6 1 2 3 ) ! ( & ! # ! An − 1 4# 4 % 2 ) 15' ( 6 2 /,+ & *., 0 *** 6 7'& .,/, %- ) 8 0 ! ! Φd (a) ) ! 2 ' - 9+ :;; < :;/ %.,,=

*=;

" l ! p l > ! p % ? " > 10l − 1 1/p - & ! # 1 0 9 ) *.* @ Rl 1 > ! 1 0 ) # & " @ % ' Rl 2 < l < 8 ' > #0 & %1 Rl ' AB l ' ! @

! p ! " # 100 ! " $" p1 $" p1 %!

&' 0

&' 8 " "! $ (" ! ) * + , ( " $" - .) ) " $" 17 &' 142857 / 0# & 142 + 857 =? 1 1 1 1 2 13 ! 17 ! 9091 #3 4 1 ) - - " $" 5 l = 2k $" p &' 2 " 0# A B # k ! " A + B = 10k − 1 (5 6 ! p / a · 10k + B ∈ N! $" 10k + 1 " + 10k − 1 A+B A+B ∈ N! k = 1 10k − 1 10 − 1 ) * 4975 4976

7" 1

$" 9949 ,

) 8 2 + & 142857 & $" 1 6 " + " &' *

, # ( ,). ) ( ) " #" ! 588235294117647 + 1 16 2 9 - 2 ) 8 & " 7 * 2, 3, 4, 5 6 & " 7 " ! * &' :& ; 142857 < + " . < 2 & & = 7 ! "

& ! 7 + " & 7 - " " /! & 142857 > 7 & 2 "3 & : ; & *# & 7 ! "3 - / ? 8 2 " ! & "3 7 " - "

& 8 2 " & 9 * & ! / " , * / # ,

! ! " #$ % x2 = a mod p ! ! 5 # & 11 ' (

p a = 0 mod p a

p−1 2

= 1 mod p

) a = 0 mod p # & * ! x ∈ Z/pZ p−1 x2 = a * a 2 = xp−1 = 1 + , & ! -. / p−1

a 2 = 1 ! / b ! s s(p−1) bs = a b 2 = 1 * b " p (b) = p−1 (p − 1) 0 s · p−1 2 * 1 s * s = 2k a = (bk )2 " # & 2 2 / 3- 4 " ! a # & p /" ! 5 $ (

(−1) p p = 1 mod 4 2 p p = ±1 mod 8 * ' $ *

! /(

2 · 4 · · . . . · (p − 3) · (p − 1) = 2

p−1 2

·

p−1 !. 2

p − 1 = −1 mod p p − 3 = −3 mod p p−1 p−k > −k k < p−1 2 k 2 ! "#$ %&

2 · 4 · . . . · (p − 3) · (p − 1) = 2 · 4 · . . . · (−3) · (−1) = (−1)1 · 2 · (−1)2 · (−1)3 · 3 · . . . · (−1) p−1 p−1 ! = (−1)1+2+...+ 2 · 2 p2 −1 p−1 = (−1) 8 · ! 2 p−1

p2 −1

p−1 2

·

p−1 2

# 2 2 = (−1) 8 mod p p = ±1 mod 8 # # '

p2 − 1 8 2

(

( ) *

+'( , -. / x2 ≡ p mod q -. / x2 ≡ q mod p "p, q > 2 '% 0 0 1#$ / ! 2 & 2 3 p p 4 / a ∈ N #5 (& +1 a ) + p a := " % p −1 a ) + p

p

a ≡ b mod p

a, b ∈ N p

b a = p p

a·b a a = · p p p

6 #

p p−1 p+1 U (p) = 1, . . . ,

O(p) = ,...,p − 1 2 2

! " 2 x ∈ U (p) 2

r(2 · x) ∈ O(p) p − r(2 · x) ∈ U (p) a p ! "

a = 3 p = 17

# U (p) ! ! 17 3 !

$

% # & # x >= 17+1 17 − x ' ( 2 )

' # ! ' $

1 2 3 4 5 6 7 8 3 6 9 12 15 1 4 7 3 6 1 5 2 1 4 7

8! = 38 · 8! · (−1)3 mod 17. ⇐⇒ −1 = 38 mod 17

! 3 * + ! ! 17 ' ,

( ' " !$

fa : U (p) x →

fa : U (p) → U (p) x∈U (p)

x

x=

r(a · x) ∈ U (p)

r(a · x) ∈ O(p)

r(a · x) p − r(a · x)

x∈U (p)

fa (x) =

p−1 ! mod p. 2

U (p)

x∈U (p)

fa (x) = fa (y) - r(a · x) = r(a · y) ' a · (x − y) p ' p ,! x − y ' . x, y ∈ Z/pZ . x = y / 0 r(a · x) = p − r(a · y) ' x + y p ' . x, y ∈ U (p) 1 ' fa & , 2 & ,2

p

p μ(a, p) = |{x|r(a · x) ∈ O(p)}|

a

a = (−1)μ(a,p) p ⎛ ⎞ ⎛ ⎞ p−1 a 2 ⎝ x⎠ = ⎝ r(a · x)⎠ mod p x∈U (p)

⎛ =⎝

x∈U (p)

⎞ ⎛ r(a · x)⎠ · ⎝

r(ax)∈U (p)

⎞

r(a · x)⎠

r(ax)∈O(p)

r(a · x) p − r(a · x) = fa (x) ⎞ ⎞ ⎛ ⎛ p−1 fa (x)⎠ mod p a 2 ⎝ x⎠ = ·(−1)μ(a,p) · ⎝ x∈U (p)

⎛ = (−1)μ(a,p) · ⎝

x∈U (p)

⎞

x⎠ mod p

x∈U (p)

a

p−1 2

2

= (−1)μ(a,p) !

" 3 p p = ±1 mod 12 p = 11 ! 52 ≡ 25 ≡ 3 mod 11 " p = 13 : 42 ≡ 16 ≡ p−1 3 mod 13 x ∈ U (p) 3 · x ≤ 3 · p−1 2 < p + 2 # p+1 3 · U (p) ⊂ {1, . . . , p−1 2 } ∪ { 2 , . . . , p − 1} ∪ {p + 1, . . . , p +

p−1 2 }

# 3·x = p $ 3 p " % " 3·x ∈ { p+1 2 , . . . , p− 1} r(3 · x) ∈ O(p) # & x ∈ U (p) ' " p+1 ≤ 3 · x ≤ p − 1 ⇐⇒ 2

p + 1 ≤ 6 · x ≤ 2p − 2

p p = 12 · n + k k ∈ {1, 5, 7, 11} 12n + k + 1 ≤ 6x ≤ 24n + 2k − 2 ⇐⇒ 2n +

k−1 k+1 ≤ x ≤ 4n + 6 3

x ∈ N 2n + 1 ≤ x ≤ 4n 2n 3 p ! 2n + 1 ≤ x ≤ 4n + 1 2n + 1 3 p " 2n + 2 ≤ x ≤ 4n + 2 2n + 1 # 3 p $ %

2n + 2 ≤ x ≤ 4n + 3 & $ ' ( 2

)*+ , - ' ( p = 13 q = 19 )** '. / , - & 0 . 2 1 2 p )*! 5 1 2 p 0 . p = ±1 mod 10 )*3 % . p p − 3 1 2 4 )*" 5 7 1 2 6 7 % 8 · k + 7 6 7 % 12n + 11 # 89$ 2 0 ::;0 # !+<6

! "!# x2 + xy − y 2 $ % &' ! (!#)! ! (!# ! "!# 5n ± 2 ! # ! # *+, !)## $

p, q

p−1 q−1 p q = (−1) 2 · 2 q p

-.$./

p q = (−1)μ(p,q)+μ(q,p) q p

μ(p, q) + μ(q, p) mod 2 x ∈ U (p) r(q · x) ∈ O(p) x p−1 }

q · x < q · p2 = 2q · p μ(q, p) x ∈ {1, 2, . . . , 2

x y ∈ U (q)

(y − 1) · p +

p+1 ≤q·x≤y·p−1 2 1−p ≤ q · x − p · y ≤ −1 2

!"#

$% (x|y) ∈ U (p)×U (q)& '

" ( ' p ) * q

& ( y ∈ U (q) r(p · y) ∈ O(q)

(x − 1) · q +

q+1 ≤p·y ≤x·q−1 2 q−1 1≤q·x−p·y ≤ 2

(q−1) 2

y=

q p

(p−1) 2

"

*+ U (23) × U (19)

·x−

q−1 2p

μ(p, q) x · q − y · p = 0 x · q = y · p p q q y y ≤ q−1 2 !

A := {(x, y)|x ∈ U (p), y ∈ U (q); −

q−1 p−1 ≤x·q−y·p≤ } 2 2

"#

%$ |A| = μ(q, p) + μ(p, q) A &' ( ' (

) R = U (p) × U (q) ' A

q−1 } 2 q q−1 = {(x, y)|y < · x − } p 2p p−1 }. C : = {(x, y)|x · q − y · p < − 2

B : = {(x, y)|x · q − y · p >

*+ B C , ' q+1 *+ ( p+1 4 , 4 ) - ' ) . ' / . , ,, ' φ(x, y) = ( p+1 2 − − y) ,0 ,, '

) B x, q+1 2

C ,', ' , B C ' q−1 '+ |A| = |R| − 2|B| 1 |A| mod 2 = p−1 2 · 2 ' 2

p q

=

q p

p, q 4n + 1

4n + 3

pq q

p

= (−1)

p−1 q−1 2 2

p q

=−

q p

= 1 ', * '

- ' ( ' −1 2 p q q p ''' 3 , * + a 313 56 4 2 7 257 313 = 257 = 257 = 257 · 257 · 257 = 257 5 2 = = = −1 7 7 5

4

) ' * 5' 67$ - 8*9 , 52 ( 1 ( 11 : , !

! " # $ " % & ' # & # ( ) ! * $ * #+ " $ " " $ , # $ " #' # - . / " 0 # 1 $ * % 2 # 3 ' 4 5 6 - " 7 $ "58 - 0 4 9 " 9" p5 / : 4 ! #+ ; <"" " "4 !

; " < !4 % &= >?@@@A " $ " 0 # . −1 p = 1 mod 4 p x2 = −1 mod p 2 p−1 ! = −1 mod p !" # $ p = 1 mod 4 # 2 % # & ' &

( x2 = −1 mod 61 )* & + , - !"# $ Z/pZ\{0} & . " / 0' !" 1 n 23 45 Fn = 22 + 1 & *6 & ( 0n ≥ 3

5 # $ p (* & Fn 0n ≥ 3 2 p

7 (* & Fn & 2n+2 · k + 1 8 1 & % ( ' ( 9 * ' (* & 264 + 1 2:

' p = 2n + 1 0n 5"9 ( 0 4( 5 # 8 3 (&" modp

5 %* 0 # ; n = 64 k−1 n = 128 p = 2n + 1 ( ! 32 mod p n = 2k

p p a ∈ N r(a · x) ∈ U (p) r(a · x) fa : U (p) x → p − r(a · x) r(a · x) ∈ O(p)

p−1 2

i=1

fa (x) =

p2 − 1 8

μ(a, p) = |{x|r(a · x) ∈ O(p)}|

r(ax) +

r(ax)∈U(p)

r(ax)∈O(p)

r(ax) ≡

p2 − 1 + μ(a, p) mod 2. 8

! p−1

2

p2 − 1 xa (a − 1) mod 2 ≡ μ(a, b). + p 8 x=1

"# a p p−1 2

a p

= (−1)m m =

ia . i=1 p

$ a, b ! % a−1

b−1

2 2

a−1 b−1 i·b j·a + = · . a b 2 2 i=1 j=1

" & '( ) *

"# ! p ! +, -! x3 = 1 Z/pZ. "# ! p -! x2 −3x−1 = 0 +, Z/pZ. ! "# ! p -! x3 − 3x − 1 = 0 !/ +, Z/pZ.

0

"# ! p 7 1 ! '. "# ! p 6 1 ! '.

2 p = 3 + 8k q = 1 + 4k 3k ∈ N 2 p

p = 8k − 1 q = 4k − 1 −2

p q p = 2q + 1 q 2 −2 ! "#$ % X 4 = −1 mod p & p = 1 mod 8 X 4 = −4 mod p & p = 1 mod 4 '( X 4 + 4 ) * + "#, - . 2 ) / p p = ±1 mod 8 %0 2 ) / p = 3 - 1 = ±3 mod 8 2 2 ) ! / 3 + - p x, q ∈ N q < p q x2 − 2 = q · p - 1 + q + ±1 mod 84 * x2 − 2 = ±3 mod 8 x 5 ' + 6 4 "#7 6 & 8 1+ + ( x2 − 2y 2 = −t2 9& x, y ∈ Z 1 t + ±1 mod 8 : 9& + 2 t x2 +(x+t)2 = y 2 x, y ∈ Z ggT(x, t) = 1 ; % 2 t 9& < t = ±1 mod 8 - t + 4 "#=

- −1 ) / p p = 1 mod 4 ( !> ! . p = 1 mod 4 2 a, b a2 + b2 = p 3 * ? 1659 % @ A -0 0 < x < p x2 = −1 mod p L = {(a, b) ∈ Z × Z|ax = b mod p} 'L 9 B * 2 p = 5 0

(0, 0), (1, x), (1, p+x) (0, p) ! A = p " # $ % & (a, b) ≤ 2 · πp < 2p ' ( ) * &+ a% b#

0 < a2 + b2 < 2p

a2 + b2 = 0 mod p ,- # (a, b) L

a2 + b2 = p. & % " !. !' / # - 0 % " 1. 1' 1 ' - % $ ' 23 4541672,6337 - $ ,489: ! 4535 ' % ;& 625 < = 1890 & & $ + &+ > " ? , @ >$ < ? + ' A 0 + % @B p = x2 + y 2 693 0 p = 1 mod 4 % ;& 625 @B# p = a2 + b 2 0 % # (∗) p = a2 + b2 = x2 + y 2

0 ( > ?C (∗)

p2 = (ax ± by)2 + (ay ∓ bx)2

p (a2 + b2 )y 2 − (x2 + y 2 )b2 = (ay − bx)(ay + bx). @ u2 + v2 = 1 & % p & B ; 0 ' . / 1640 &+ ! % " !. !' BD % % ; & 4E23 " (μ, ν) ∈ N × N % # % μ · ν & " % & 0

μ · a2 + ν · b 2

! "

#

$

% & # '

(

) * +

Z[i]

-. + ',/ %

' -#

",

Q(i)

'

+ /

01 % # ' ' + . 2

3

! " 4 ' ' 5 6

3

. 2

# $ % '&()

07

5 (8

% & (

! " # $% " # &' & ( ) ' $* + '$ , ' '- ' + " . '

- / / 0) ' 0 1 " ( , 2 ( " 3 45 ' ' - - + ) - 5 n > 2 1 xn + y n = z n . 0 " ( 2 - "6 / ' 7 8 - 1 / - 2

400 ! "# $ % & ' () *+ , - '. / , (a, b, c) & &. 0&$ a2 + b2 = c2 1

(a, b, c) a b ! a (a, b, c) p q q > p a = 2p · q b = q 2 − p2 c = q 2 + p2 " p q # p > q a = 2 ·p·q b = p2 − q 2 c = q 2 + p2 $ (a, b, c) % b = 7 p & ! $ ' (a, p, c) # % (a, 81, c)

% (a, pn , c) $ p & !

a 4 c 3 (a, b, c) c 5 ( a b 3 ) (a, b, c) c 5 a b 5 " & ( #* a ≤ 100 ' #!

&. , 2 ' 3& 45 a ∈ [0, 100] b ∈ [0, 75] 6 ' 7 $ c2 = a2 + b2 , & (a, b, c) &. 8 , & & $ 9 : ;

pp p p p

p p

p

p

p p

p

p p p p p pp p p p p p p p p p p p p p p p p p p p

p p p p p p pp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

! " # $ "% & ' " ( ( % ) a2 +b2 = c2 " *+ , a3 + b3 = c3 an + bn = cn - , $ *+ . # ( / '( (' $ n = 4 / & / (' n = 3 / " / 0 1( + 23. 2 1993 , 3 " & (' 4 / " $ $ , 5 & 2 , 6 $ / / $7 . . /% )$ 8 , $ . 0 9 : ;< =( > ??7 > @ 2 A $7 # ;B9 2 + - 0.70 C ;D7 > 2 1 8 $" & 0 B9: B< >7>B

" $ (' $ n = 4 / E 0 / " , " A F >? 0 / G 0 ) $ : B< 2 D; F / (' n > 3 ) xn + y n = z n " ( :"< // *+ . $/+ ) ( / # , ( / , $ "

, / ( / "

x > y ggT (x, y) = 1 p > 2 ggT (x + y, xp−1 − y · xp−2 + . . . + y p−1 ) p q #( +H 6 x = −y mod q xp−1 − y · xp−2 + y 2 · xp−3 − . . . + y p−1 = p · xp−1 = 0 mod q. , q = p ' q 6 " x I q (x + y) ' ( q 6 " y H q ggT (x, y) , ggT (x, y) = 1 q = p 2

p

2p + 1

xp + y p + z p = 0

(∗)

(x, y, z)

p

x·y·z

x, y, z p x · y · z q = 2p + 1 x y z −xp = (y + z) · (z p−1 − z p−2 · y + . . . + y p−1 ).

s ! " s = p p xyz # (y + z) = Ap ,

z p−1 − z p−2 · y . . . + y p−1 = T p

(x + y) = B p x + z = C p $ p = 21 (q − 1) q %!# q x · y · z $ xq−1 = 1 = yq−1 = zq−1 mod q " ! & ' q = 2p + 1 ( ) (∗) q # xp + y p + z p = x

q−1 2

+y

q−1 2

+z

q−1 2

= (±1) + (±1) + (±1) = 0 mod q

q ≥ 5 *!# q (x · y · z) q x $ B p + C p − Ap = (x + y) + (x + z) − (y + z) = 2 · x B p + C p − Ap = 0 mod q + ! ,-# q (A · B · C) q x q (B · C) q . B B p = x + y y +/- q z x y z ) q C q A # 0 = Ap = y + z mod q −y = z mod q T p = p · yp−1 mod q 0 y = B p mod q + q x- T p = p · (B p )p−1

' q B p = ±1 mod q p−1 # ±1 = T p = p mod q 1 p = ±1 mod q 1 < p < q − 1 2 2 "3 ) " %4%556 7 8 & 9 $ !

! " ! #

$ % & ' ( )* % ) * % & ! + % "

! " #$

% "! # &

!

#! ' ## (

' ,-'. /-0 * % )*. % 1 ) +"

! * # #

,

#

- .!/

! #

* 0 # )

1

' # 2 & !

" )

3 4 / - # " ## ) # 1 3 .

3 ! .5 # / . " .

" 1

"

'! . 0 # 1 # " &

# # #! , 3

3 3 . .. (

( p% 2 · p + 1 % ( ( " ! 2. 33 4 3, 11, 23 . . . 5. " ( % 6 7 ! * '+ +8 % . % .! 9 * :; <<:% /- =

9 *& :> 5? @#6 @ = & '. % !6 % '. & A ) . B ) 6 ). !

"

"

"

! "#

"$

%

& ' = {0, . . . , 40} ( ) *+* , ' - ./ 0

C:

n → 7 · n − 4 mod 41 ∈

1 C(n) = c = 7 · n − 4 mod 41 2

n 34 ) 5 / 6

6 / 7 8'7 7 9 C 2 4 :7 7 7 * ) ; :' 6 < .+ = >; ' ?@ = 9 ) ( ' - ' ' - 7 & 9 * - 0 40 & >6A- @ e 40 ' B' - 7C / n = 0, . . . , 40 ./ 9 0

V :

n → c = n7 mod 41 ∈

.

' ./ & )8 '/ 41=? 9 13 30 ' - '4 ? ' ' D ' 7'E ; '' -' 1 ' - 137 = 26 mod 41 307 = 6 mod 41 ) , )8 E F ./

+ ( / 6'- * , cG & 6 / 7 2 G & ' ; ' p 9 & 6A- d V (n)d = (n7 )d = n mod 41 / n ∈ B n40 = 1 mod 41 / b ∈

$ ) / k ∈ Z n40k = 1 mod 41 B/ n ∈ C 8 n40k+1 = n mod 41 & d ∈ N

7 · d = 40k + 1 / k ∈ Z

& ' ./ A- 40 '

) 7 9 ' 40 2 4+ & H I4 7 ' '

p

7 · 23 = 161 = 1 mod 40.

E:

c → c23 mod 41 ∈

E(V (n)) = E(n7 mod 41) = n7·23 mod 41 = n mod 41

!"# $ % e = 17 # &p = 41 $ ' ( p = 41# e = 11 = d &% ) * + '' '' p = 41 e# % ) * + '' &d = e &"$ , +

- .' % / 0 11 $

# $ $ 2 3 244) + * 5 6 255 '7 3 ' 8 # 8 .+ . 3 # 1# 9 # :'* ' 32 255 − 32 = 223 3 223 1 $ 0 '+ +' 19 ' + 222 17 *

'; 6 , V : {0, . . . , 223} n → n19 mod 223 ∈ {0, . . . , 223}

' " 6 % ''# $ 6 . ( 6 2* + ( & + 7 1 6 '7 % <

= % # % >1 < = % V # V ◦ V < = % # ' 2"2 ?"? $

<

p

!

@ . 3 6 2, . . . , 10 $

'/; c = ne mod 11 6 p = 11 1 2;

' $ ; ' # 3 2 3 7 $

% ) e +$ d<

2e = 7 mod 11 ! " #$% & '( ) % * 2e = 7 + e ) $ 7 ' 2 , log2 7 - ! #' 10& log10 x = lg x ) #' e = 2, 71828 . . . & loge x = ln(x) % ! "% . $ /

" ax = b lg b #a, b 0- +1& * x = 2 . lg a # 3 + a, b& p / ! 4 ., 5 " ! ' 2x = 3 mod 17 % " ax = b mod p, b = 0 mod p a $, $ p 6 . , 4 '( , 7 89 / p a $, b ∈ Z/pZ ! i ∈ {0, . . . , p − 2} ai = b ! + $ b ' a , inda (b) , p $4 $ 6 : 4 " , : )

p a

inda (b · c) = inda (b) + inda (c) mod (p − 1)

inda (bc ) = c · inda (b) mod (p − 1)

inda (1) = 0 inda (a) = 1

', , ; 6 . / ' 3 $, $ 17 * a ind3 (a)

1 0

2 14

3 1

4 12

5 5

6 15

7 11

8 10

9 2

10 3

11 7

12 13

13 4

14 9

p

6 13 17 ! " # $ 315 = 6 mod 17 34 = 13 mod 17 "% 6e = 13 $ 315·e = 34 mod 17 315e−4 = 1 mod 17 3 & '$ 15e − 4 = 0 mod 16 15e = 4 mod 16

e = 12 mod 16 ( ) 612 = 13 mod 17 "% *

& + ! 16 6 13 ' 15 16 , ! 9 6 - ./ 0 1 & a 2 p % b c $ b = aind(b) 3 c = aind(c) be = c ae·ind(b) = aind(c) e · ind(b) = ind(c) mod (p − 1) 1 ggT(ind(b), p − 1) # ind(c) / , e " ' 4 / e ' %$

• + z ind(c) = ggT(ind(b), p − 1) · z • , 5&6ind(b), p − 1, x, y, ggT) 6 7 8 9 : & ; x, y x · ind(b) + y · (p − 1) = ggT(ind(b), p − 1) (z · x) ind(b) + y · z(p − 1) = ind(c)

• c · x / e + ! & 5 % & ! 8 8 " & & 4 5 + & & + + & + & & 4 8 & ; 1 & 2 2 % & < 8 & % b c + = "' $ 7

p = 11 p = 17 ! p

a " #$% " & ' a mod p! b ( )' *+ , 6x = 4 mod 11! , 9y = −2 mod 17 - . +/0 1 & 2 p (a) =

p−1 ggT(ind(a), p − 1)

3 p = 8963 4 < p 5 $ c = n143 mod 8963 , * & 6 7 8 * 6 9 4 5885 8

0 :# 2 , , - ;

" 8963 *+ d e * ! 4 4701 4 8720 " < *+ d e ! * ! p ! d e :2 ,

* p − 1 = 8962 = 2 · 4481 ==3 ! + >? > 8 5 5! )' ? $

ax = b mod q ;9 " @ + & + + > ? >

! p d "! e# $ % & ! '

( % ) *

$ ' $ "+, - #% & . ! *

$ / $ %

a ∈ Z/nZ Z/nZ

ggT(a, n) = 1

! Z/pZ p " #

p

$ % Z/9Z 1, 2, 4, 5, 7, 8 & 6 ' ( #

'

ϕ)(

p " (* +

"

& " , - " $ $ 9% " 2 9 . 1, 2, 4, 8, 7, 5 - . #

"

/ 0 '$

1

2 -

(

1

m ∈ N

Z/mZ

1

a

1

a

- Z/mZ 3 $$ $$ + 45 $$ Z/mZ 6 7( x Z/mZ E = {xn | n ∈ N} 6 . x " x & 8 . x

3 $$

- . p " $$ Z/pZ 7(

" 3

1 Z/9Z $$ 7( 9

1

:

* 1 #1 . #1 .

2 Z/27Z 2 Z/81Z 2 Z/3n Z

7 (2) = 3 49 (2)

7n (2) = 3 · 7n−1 n

≤ 100000! 2

" # p! 2 $ % Z/pZ! % Z/p2 Z 1093 ! ! & ' &

6 52 · a − 1 a ∈ N 28

( 5a − 1 128 7 ! 97 )

) & b 2b − 1 97 * ( 2300 · a − 1 97 ! 257

b ! 5b − 1 257

) n ∈ N, n ≥ 2 2n |(52

p xp−1 = 1 mod p2

n−2

− 1)

Z/pZ

x

Z/pZ x ∈ Z x xp−1 = 1 mod p2 xp−1 = 1 mod p2 ! "

y = x+p p−3 · x Z/p2 Z # (x + p)p−1 = xp−1 + (p − 1) · xp−2 · p + p−1 · p2 . . . 2 p−2 2 p−2 2 = 1 mod p !$ $" 1 − p · x = 1 Z/p Z $ = 1−p·x −p · xp−2 = 0 $ −p = −p · xp−1 = −p · xp−2 · x = 0 mod p2 xp−1 = 1 mod p2 −p = 0 mod p2 2

Z/pZ

p

Z/p2 Z

% & ' " Z/pZ xp−1 = 1 mod p2 % d = p2 (x) ( (p − 1) · p ) " xd = 1 mod p (p−1) d d = (p−1)·k * d ( (p − 1) · p k = p k = 1 "+ xp−1 = 1 mod p2 $ 2

Z/p2 Z

p

r ≥ 2 xϕ(p

ϕ

r−1 )

= 1 mod pr .

x

r r = 2

r ≥ 2 r−1 r−1 !" # xϕ(p ) = 1 mod pr−1 $# xϕ(p ) = 1 + n · pr−1 % & p n # # ' r xϕ(p ) = (1 + n · pr−1 )p

= 1+p·n·p

r−1

p + · n2 · (pr−1 )2 + . . . 2

r xϕ(p ) = 1 + n · pr mod (pr+1 )

= 1 mod (pr+1) , p ( # n

2

p r ≥ 1 Z/pr Z x Z/pZ xp−1 = 1 mod p2 x Z/pr Z r ≥ 1 " & ) # x # p %

xp−1 = 1 mod p2 d = (x) mod (pr ) xd = 1 mod pr $# xd = 1 mod p% (p − 1) ( # d $ #' d ( # ϕ(pr ) d # ( # (p − 1) · pr−1 ) #' d = (p − 1) · pk−1 = ϕ(pk ) r−1 1 < k ≤ r )* k < s% * xϕ(p ) = 1 mod pr % ) 2 & +&# $# d = ϕ(pr )

x d mod p xd = 1 mod p2 r−1 r−2 d · r ≥ 2! x p = 1 mod pr xd · p = 1 mod pr

r−1 , * ' xd · p = 1 mod pr ## r ≥ 2 ) d r = 1 x = 1 mod p

r r p xd · p = (1 + n · pr )p = 1 + p · (n · pr ) + · (n · pr )2 + . . . 2

= 1 + n · pr+1 + . . . = 1 mod pr+1 . ( # #

r = 2 r−2 ' # r ≥ 2 xd · p = 1 mod pr−1

r−2 # xd · p = 1 + n · pr−1 % &

p

n

xd · p

r−1

= (1 + n · pr−1 )p = 1+p·n·p

r−1

p + · n2 (pr−2 )2 + . . . 2

= 1 + n · pr = 1 mod pr+1 .

p x d mod p xd = 1 mod p2 (x) = d · pr−1 r ≥ 1 pr r = 1 xd = 1 mod p2 xd·p = 1 mod p2 s = (x) mod p2 s d · p xs = 1 mod p d s ! s = d · y d · p = s · k = d · y · k p = yk y = 1 k = 1 k = 1 !

p = k " s = d # xd = 1 mod p2 $

% r ≥ 2 & ' ( ' r−1 r + 1 '" xd·p = 1 mod pr+1 r xd·p = 1 mod pr+1 % ) s = (x) mod pr+1 s *+ d · pr s $ s · y = d · pr !# xs = 1 mod pr s = d · pr−1 · k k · y = p %

' # , k = p 2 !

10 487 4872. 14 29 292 (2) = 10 · 11r−1 ! r ≥ 1 (2) mod 17r (2) mod 13r

"

#$ % xb = 3a + 1 #$ % xb = 5a + 1 p #$ % xb = pa + 1 & % ' xb = y a + 1 $ y ( ) 5b = 6a + 1 #$ 7b = 6a + 1 #$ 13b = 6a + 1 17b = 6a + 1

#$ xb = 6a + 1

p 2p−1 = 1 mod p2 x, y, z ! p " xp +y p +z p = 0 2p−1 = 1 mod p2 #$ % & $ ' #( $ ) * * $ ( " $+ , ' ( % ! - & +- ( .+ 1000000 % " , p 2p−1 = 1 mod p2 / ap−1 − 1

* 01 % a 23+! p p #a ≥ 2' ( * 0 1 ( % 2 3 5 100000 * 01 4 p 5 4 ) , ! p x, y, z " 6 p xp + y p + z p = 0 qp (l) = 0 mod p - l ≤ 31 #6

7 8 * 9 : # p' 7 4 5 " #;;' <0 '

qp (a) =

! "! 2 × 2 # Z/nZ $ !

"!

n

n n−i i n n n n n−1 n−2 n a (a + b) = a + ·a ·b+ ·a · b + ... + b = ·b . i 1 2 i=0

% & '( ) * + & "! $ * kp = 0 Z/pZ , ! p 1 ≤ k < p $

(x + y)p = xp + y p x, y ∈ Z/pZ & ρ : Z/Zx → xp ∈ Z/pZ !-! . (1 + x)p = 1 + xp &"! . /

0 xp = x x ∈ Z/pZ 1 2 /

- "! - % 3 "! (a+b)2 ! a2 +ab+ba+b2 . 4

"! a2 + 2ab + b2 *

ab = ba

a, b ∈ R "! * R / . . A ∈ R(2,2) # 5 R / * R[A] / R(2,2) * A !6 * R[A] / 4 "! / 7 6"!

B(n, 0) := B(n, n) := 1 n ∈ N B(n + 1, k) := B(n, k − 1) + B(n, k) 1 ≤ k < n

1 1 1 1 1

! " # $ ! % Z ! "

1 2 1 3 3 1 4 6 4 1

(a + b)n =

n

R

a, b

B(n, k)an−k bk .

k=0

& ! n = 0, 1, 2 ' ! n + 1 ( (a + b)n · (a + b) % ' ) an−k bk ! B(n, k−1)+B(n, k) = B(n+1, k) ! " !2 * ' Z + , -

0 < k < n k! · B(n, k) = n · (n − 1) · · · (n − k + 1)

& n = 0 n = 1, 2 " ! " ! ! n ≥ 2 B(n + 1, k) 1 ≤ k ≤ n % k = n. B(n + 1, n) = B(n, n) + B(n − 1) = n + 1 */ n!(n + 1) = (n + 1) · 22 & k < n !/ k!B(n + 1, k) = k!(B(n, k) + B(n, k − 1)) = (n + 1) · · · (n + 1 − (k − 1)) ! 2 0 &! !. ! /

B(p, k) = 0

R

R

p

1 ≤ k < p

p ( , k! p 1 B(p, k) 2

ρ : R x → xp

p

!

(x · y)p = xp · y p 1p = 1 (x + y)p = xp + y p B(n, k) = 0 1 ≤ k < p 2 ! α " Z/pZ(2,2)

#$ % x2 + p ·x+ q R = Z/pZ[α] $ $

& "

(a + b · α)p '

& αp

#

( #

)

R

α

αp = 3

p = 5

5

!

"

5

!

p > 2

x2 − x − 1

p

p

αp = a αp = 1 − α

α2 − α − 1 = 0 R ( (2α − 1)2 = 4(α2 − α − 1) + 5 = 5 p−1 p−1 (2α − 1)p−1 = ((2α − 1)2 ) 2 (2α − 1)p = 5 2 (2α − p−1 1) = 2p αp − 1p = 5 2 (2α − 1) '

* p−1 2 · αp = 1 + 5 2 (2α − 1) p = 5 (

2α5 = 1 α5 = 3 + 5 , " p (

5 2 · αp = 1 + 2α − 1 = 2α ( αp = a 5 - (

5 2 − 2α (

αp = 1 − α

bα ∈ R, a, b ∈ Z

p = 5

5

"

5 = β

2 βp

= 1 R

= −1 ∈ R 2·αp = 1−(2α−1) = 2

# $ % &'

β = a+

p−1 2

p−1 2

β 5 = a + 3b

!

p

!

(

p

βp = β

β p = a + b(1 − α)

R 2n =

n

B(n, i).

i=0

n

i=0

B(n, i)2 =

2n n

!"#

!" $ %& '&( )%& '&(* ++ * ,- ./ 0 " "

" , . 1 - 2 a, b R * 3

a2 + 2ab + b2 = (a + b)2

4 !" B(n, k) Z/2Z !" n = 16 B(n, k) Z/2Z 0 % 52 !" n B(n, 2) = 06 52 !" n ∈ N B(n, k) = 06 52" !" $" !" 7

Z/3Z, Z/5Z

8 9 K ,#- (Q, R, Z/pZ)* : !" 1!" x2 + ax + b = 0 ;a, b ∈ K <# " 4 R = K (2,2) 2 × 2 % 2 K 0 1 R <# 1!" 0 %= α = −b −a α $ K[α] ! β = b · α2 <# 1!" 0 ρ : K[α] x + yα → x + yβ ∈ K[α] " 7

-" * > !" 9 K > 0 ? -" > 0 N : K[α] γ → γ · ρ(γ) -$ K[α] ,#- 0 "@ 9 = 0 2!" %7 - $ K[α] 9

K = Z/pZ !" αp K[α]

! "# $%&'() * %+,-

! " # $ % & % ' & ($ ) !$ ! # *+ , . /0 1 # F7 = 22 + 1 1 /023 ($ # 4 F7 = 59649589127497217 · 574689200685129054721 5, ! /02/6 $ % . 4 ) p ap = a mod p # a ) # a p 4 ) # an = a mod n n 2 & 7 8 97: a = 24 7

/ =

6 5 ; <6 23363148097 = 131072(= 221 ) mod 3363148097 # > 5 =/=6 Rn = 19 · (10n − 1) = 111 . . . 11 5n ?6 n 3 13 2R 8 Rn @ rn 24 26 = 4 mod 6

n

n rn

n rn

3 8

4 937

10 242935453

5 9961

6 42869

11 2992649798

7 1107782

8 8230414

12 34901278238

9 96666315

13 920227682634

Rn

n

Rn Rp p

! " # R3 , . . . , R13

R16 R2 $

R2 , R19 , R23 , R317 R1031 ! % &' p < 10000

Rp & ( ) !

* + # Rn &' n < 100 ! " ' , &

- ./ & 01 ' & -& Rn n

2n 3n n !" " # 1, . . . , 31 $ %& ! '( ) '&* 31 ) " + ,"! 5099719 86146913 -

. #) R9 R10 R12 ! .

!) ) R11 11111111111 = 21649 · 513239. " /0 12 " ) 3! #) ) )! )) " 4 5 6 7 8 9 :; ) # 11 111 111111 3 < (!) 9 !) ) R13 !") 4 ) ! = 9 ) 9!) 8 ! > #) ) ! ? ' 100 ) . 4 @ ! 8 ) R13 !

. #) R15 A) ?2906161 B. -C 9 # ' Rn = 19 · (10n − 1) ) an − 1 ' ? / 0 a. a−1

Dp = 12 · (3p − 1) p ! " # D3 # D7 # D13 # D71 # D103 D541 $ % % Dp # & ' " Fp a = 5 Ep a = 11 '( ! ) # F3 # F7 # F11 # F13 # F47 # F127 # F149 # F181 # F619 # F929 # E17 # E19 # E73 # E139 # E907 ( ! * + ,

2n ± 1

13 · (4n − 1)

! -. / 333 . . . 331 zn = 31 (10n − 7) 0 # 31, 331, 3331 & ! # / + + ! a = 2 1 n )

2 8 2

9 235425188

10 2799910860

11 1684575087

12 38750750244

2 + # z9 , z10 , z11 z12 & z5 z8 # + + 0 3 + + + " z2 z8 " 1

! 456 7

a = 3 5 z2 , z3 , z4 + ' + + 454

t > 1 8 z9 = 333333331 + # t 9 z16k+9 (k ∈ N 7 333 . . . 31 8 u > 1 z12 + # u z18k+12 (k ∈ N) '3 # zn

45.

: 0 " z9 z12 ; < t u ! " n = 1, 2, 3, . . . ' = n Sn = n4 + (n + 1)4 ) 2Sn 3Sn / Sn : n # " Sn Sn #

n4 + (n + 1)4 n ∈ N 17|Sn ! 2Sn = 217 mod Sn "# $ % % & ' ( ' a, b ∈ N ggT(a, b) = 1

" a + b · n n ∈ N ) *+ ( 1805 1859 ' )" (, % - .$ % ' / ) ( 0 % - 1" 2 2 ' ( ) ' *3 "

4 0

/ 2 $ 5# ! ( ) 4 0 (6 ! % - * $ # , ', 5 7

(" f (x) =

n

ai · X i )8 59

i=0

+ , k ∈ N n ≥ k f (n) " ) $ ) n4 +(n+1)4 %, " n # $ : 5 0 ) Sn 1 0 ) n2 + 1 , " 1" 1 % # ; # ' * - 1"! an = a mod n 2n = 2 mod n n

! "

2n = 2 mod n # n $ # % $ % & ' 2n = 2 mod n

n ( $ )*

+ ,

, # % #, - + .! / " 0 ) 1 + 2 3 #, n = 1, . . . , 99

2n mod n =?

n=1

100

99 2 !"# $ % & " '% ( % & " ' ) * " + , -% . & , 2n = 2 mod n " n / "

) 0 1 2 %" 3 " ' 4/ + 5 & , +' " " & " , & +' 6

, "" " & 7& -" 3 & " "

"# + 2 8 39 : 39 & " 1 4"$ 5 ;' *" < & 3 < 1 =" 2n = x mod n ) % " ) % 1 >

2n

Z/nZ

! " #$

n∈N

! " #$

n%

$ $#

n

2n + 1

n ∈ N n

2n + 1

$ & '()* $ #$ $)

$ +# , $ -) -))$ ./ $ &

n

)% $

2n = 13 mod n% 2n = 17 mod n% 2n = 67 mod n

0 * .*% $ $ "#

n>1

!

2n = 1 mod n

φ12#&)

/!

ggT(2a − 1, 2b − 1) = 2ggT(a,b) − 1

" $ $

n > 1 2n = 1 mod n d = ggT(φ(n), n) 2d = 1 mod n 1 < d ≤ φ(n) < n 2d = 1 mod d !

" # $!% & ! ! ! '() * *& r > 1 & n +

2n = r mod n , r n < L $L - 2n = r mod n $r ≤ n " * '(. -

/ $

2n 0 n 3 1

1 ! 2 3 1 0 4 *& / 5 * ! *&* 4 5 & !1 n = 1 $6

# 5 ! & 5 7 3 8 9 n $ ! * 24700063497 = 3 mod 4700063497 : , * 1 $; & ,* < 19 · 47 · 5263229 " ! % * 5 5 * 4 #

= $ ! > ? 2n−2 = 1 mod n * ?# @A $B(C@ 'DB<'D' * ! *

-

! * !

n 2n = 4 mod n

'(D n := 2m − 1 ! m = 4700063497 $ 2m = 3 mod m * '() 0 n 2n = 4 mod n 4! = $;! " 22 − 3 = 1 mod (2m − 1)% 0 n 1 E = ! * ! n * 2n = 4 mod n " n = 4208 ?#

!& n * < : B(C) ! ! ?# m

n 1000000 20737 93527 228727 373457 540857 2n−k = 1 mod n !

"#$! %"&%"! ' ( ) * + 7 , - . -/0 1 23 / 2n = 4 mod n 7 4 5 1 0 0 6 / n 2n = 4 mod n 0 3 7 6 0 . / -0 /0 5 8 90 8 * / + : 15 n > 6

p / 2n −1 2m −1 0 m < n * p 0 / / 2n − 1 p 1 p = 2nk + 1 ≥ 2n − 3 > n ggT(p, n) = 1, k ∈ N * n > 8 0) 2n = 4 mod n p = 2(n − 2)k + 1 / / 4(2n−2 − 1). *; - np − 2 = (n − 2)(2nk + 1) 2np−2 − 1 = 0 mod (2n−2 − 1) 2np−2 − 1 = 0 mod np 9 < ' 9 ; - + = 5 + k / 5 + n 2n = 2k mod n. * 0 * 5 a ≥ 2 2 . 90 5 0 / .

' =0 n 2n = 8 mod n 2n = 16 mod n 2n = 32 mod n 2n = 64 mod n 6 ; 19147 / . - 23 ' 2n = 5 mod n > 0 . ? ( 3 5 = 5 k ' 2n = k mod n 23 / / 23 ( ) 5 0 p @ n / p 2n = 2k mod n 5 k

k = 1, 2, 3 k

2n = 2 mod n n !" # $ " % & ' ( 2 99 ) * & + , - % ./ + 0

• ! "

# • $## % & ! #

' n #

2n = 2 mod n ! ( ) * # +

,#

+#

# -

. / /

% 0 +# 1 # 2 +# 2n (" + # % n > 99 , # 2 % n = 100, 102, 104, 105, . . . . $#3 4 $5# * , ) . / # 6 +# 2 + 2 $#3" 78. +#

1000 # 10000 # 1000 )#" # 9 ) ! # : # #3 / + ! / 1 2 ; ' " 5 :# 11111 . . . 1 # <# " (an − 1) 4n − 1 = (2n − 1) · (2n + 1) (a − 1) =

,% a = 4. 4" ' ,2 . 1111111 . . . 111 $ 4" 2 ( 5 > 1, 5, 21, 85 341, 1365, 5461 ( 21 #3 "

,5461 =?. 4 9

:% 6 n > 2 v(n) =

(4n − 1) 3

2

#9 $

(2n − 1) · (2n + 1) > v(n) = ( 3 0 :# 3 n ' 3 0 ! 6 2n − 1 2n + 1 : % n > 2> > > 1 ( 3 3 ,: > :# # > 29.

p

> 3

2v(p) = 2 mod v(p)

v(p)

! v(p) v(5) = 13 · (45 − 1) = 341 = 11 · 31 ! " # $ 2341 = (231 )11 = 231 = (210 )3 · 2 = 2 mod 11 2341 = (211 )31 = 211 = 2 mod 31# ! % & # 31 11 # '( ) ! " #* + , n = 341 ! - 2n = 2 mod n# . ( % 2p = 2 mod p# p = 2 mod 2p#

2p

2p − 1 = 1 mod 2p

p = 3

2p + 1 = 1 mod 2p. 3

$ v(p) = 1 mod 2p# 3 · v(p) = 22p − 1

22p = 1 mod v(p) . v(p) = 1 + 2pk(k ∈ N) 2v(p) = 2 · (22p )k = 2 · 1k = 2 mod v(p) # !/ 4p − 1 p > 3 " v(p) = " # 2 3

0 1# '&20 * - - n 2n = 2 mod n 3 4 " # 5 v(p) ) p = 7, 11, 13. '%! 6 " ( 3 7 %486*# & "

. & - 9 3 2n = 2 mod n# n 2n = 2 mod n 2n−1 = 1 mod n ) 2:& #

341 ! " # ; & " # 2

. <1#

561 645 n ! " # $ 341 561 645 1000 10000 22 % 1229 100000 78 1000000 245

N 103 104 105 106 107 108 109 1010

& ' ≤ N 3 22 78 245 750 2057 5597 14885

& ≤ N 167 1228 9591 78497 664578 5761454 50847533 455052510

( 4369 4371 %) ' '&! 25 · 109 1105 # 1000 " ! '

*++ , ! ! ! ! - 100000 . # # ! / 0 1 2 34+ , $ 5 % 6 7 8 161038 , ) ! ! ! .! & 8 & 161038 % & 8 2161038 = 2 mod 161038 343 %9 2m − 2 : 4 %343 44*'444 ( 2n−1 = 1 mod n ;

1951 161038 ! "# $ % $ & " ' () *+,+ -./-.+ 0 1 2 · 178481 · 154565233 2 · 1087 · 164511353 2 % 1 223 = 1 mod 178481 21119 = 1 mod 154565233 2543 = 1 mod 1087 241 = 1 mod 164511353 3 & 4 4 5 & 2n − 26 " ! n = 2, . . . , 21, . . . 7 & "# 7 3 8 9 2465794 − 2 : $ 2 6 2n = 2 mod n 100000 ! 100000"#

$ n > 2 ! n % # $ 2n = 2 mod n n % # & n "

%# $ n # & n #

' 100000(

341 1905 4033 7957 11305 15709 23001 31417 39865 49981 62745 74665 87249

100000 561 645 1105 1387 2047 2465 2701 2821 4369 4371 4681 5461 8321 8481 8911 10261 12801 13741 13747 13981 15841 16705 18705 18721 23377 25761 29341 30121 31609 31621 33153 34945 41041 41665 42799 46657 52633 55245 57421 60701 63973 65077 65281 68101 75361 80581 83333 83665 88357 88561 90751 91001

1729 3277 6601 10585 14491 19951 30889 35333 49141 60787 72885 85489 93961

! " #"$ % n&' ( ) * ! + #, -. , ! 200000000 +& '

lim

x→∞

/ < x = 0. < x

" 0

! " # $ 2n = 2 mod n2 n2 % & '

n = 1093 ( 1093 &) * +

, - . / 2p − 2 p = 1093 0 . *

! 1 &22 223 4 5 6 7

6000000000 p 2p = 2 mod p2 p = 3511 ! 8

1 ' ! / / 2 + 34 ! # " 5 & ' ! p 2p = 2 mod p2 + 6 "

7 -8.' 9" :' :8;.<' ) = ( 1093 + ' ! "> ? > > ! > ) $ 6 > + 6 " @ ! ? " 0 7 p > 2 ' "$ A xp + y p = z p

! ,5 # ! ,5 B& ' p ( ! x · y · z ' 0 2p = 2 mod p2 + # + A % )C + ? " )6 +& ? D ( E ! / ) ' #"$ & ;6 2n −1+ ? n ( ' 2n −1 ( + ? ) F )6 0 6 2p −1' p ' % )6 Mp = 2p − 1+ ) 6 (5 / #M2 , M3 , M5 & #M11 = 23 · 89) + D 0

a

2p = 1 mod (2p − 1) p p 2p − 2

(2p )

(2p −2) p

2p −1

2

= 1 mod (2p − 1) = 2 mod (2p − 1)

Mp ! n

2n − 1 ! " # $ % % # & ' ( #

!) *+, *. & - " / & 0&" 1 " "" # , 2(34 5 2 0&" ) , ) 2n − 1 6 7", 8 ) ,

$ ",

" 9 #

1) 2p − 1 : & ; #

1) & " # ,

5 , & p < #

1 ) Mq

p2

< % Mq

2p = 2 mod p2 % 0&" 2 + &8 = >

' % * p Mq p2 2p−1 − 1- & * p2 Mq - 0 * ? @ ?- " " 8 ,

p p am − 1 p2 ap−1 − 1

p2 < % am − 1

* r := p (a) m p − 1- n

A ' &8 " . Fn = 22 + 1 B . > n + 1 < 2n &8 n > 1

a " # $ % n & 2n n ' 2 ( 341 ) * ' & a > 2 % 3341 = 168 mod 341

341 ! " n 3n = 3 mod n 36 = 3 mod 6 #$ #$ % & n 3 ' ' ( 391 = 3 mod 91 ) * +, &-* . " n 3n = 3 mod n / 0 ! 3 1 3 2 & n ! -3& 3n−1 = 1 mod n ) 4 $' ) * #$ 5 0 ! a

) * + $ 6 a $ " & & 1 n

a # $ " n / 05 ! a an−1 = 1 mod n 6 ' 0 7 0 ! 2 ! 0 341 0 ! 2 ' ! 3 %# ' ( 8' ' ! 0 0 ! 5 ) 9 # 6

' 2 & ( : ( 6 n 0 ! 2 n 0 ! 3 5 - ( n ! a > 2 6 - # ) - 2 : #; ) #$ %# ' #$ 0 ! < 561 2561 = 2 mod 561 ' ! '- & !

#$ ( a561 = (a187 )3 = a187 = a · (a93 )2 = a mod 3 a561 = a mod 11 a561 = a mod 17. 6 ' ' #$ $ a ( a561 = a mod 561 6' #$ 561 # ! a a560 = 1 mod 561 % 561 0 #$ 7 ! ) * +< . " n / = 5" #$ n # ! a ( an−1 = 1 mod n. > 4 = 5" ' 100000(

a

561 15841

1105 29341

1729 41041

2465 46657

2821 52633

6601 62745

8911 63973

10585 75361

(6m + 1) · (12m + 1) · (18m + 1)! " m = 5 · 7 · 11 · 13 · . . . · 397 · 882603 · 10185

" 1057 # $ % " & '

( ) * + ,- .$/$)! ) " ( ) '

" 01 ) $2 a ! "# a $% & vp = (a2p − 1) : (a2 − 1) ! p ' a(a2 − 1) %( ) *+ ! ( a = 2 (, + - . M11 "# 2(3, 5, 7) / 0 p = 6m+ 1 q = 12m+ 1 r = 18m+ 1 " 1 p·q ·r 2 ! . 3

( !/ 4 ) / 5 4 101101 . *

, 6 7 (

N ! " p p − 1 ' N − 1 N $6 3 ! ! ) 6 3 # , !, 8 66 + 1 , 8 nn + 1 $ ( . "

9,

- 3 ) 456 1 ) ) " $ ! )

) 1057

!

3 ) ) ) 3710 " ! 7$! 8 " 3 ) ') % 8 ! 3$ 9$ :4 6! ; ; ;$ ($ < =>3 # ? 7 ) 1989/90 >

3

' " $ 3$ 9$ :: 5! 46446$ 1012 1000 3 3 ' ! 2102 3 4! 3156 5! 1713 6! 260 7! 7 8 ) 3

8 ' $

n b n ! " ! # n ! # $! % ! &' ( n ) " ! " ! ! ( * n + " ! n b0 # bn−1 = 1 mod n ' , 0 ! ! # - n ! # ' . ! /0 n + " ! # n ! 50 % 1 ! n ! 1 n 2 0 n = 341 = 11 · 31 # ! - b = 3 - . b ! 3 ! % 31 % 11 4 ! 1 340 30 + 10 = 40 3 ! % 11 31 # 341 + )

2 - 340−40 = 150 2 ' . 0 5

" + 300 ) 6 + 5 / 150 7 8 4 9 : ' ( 0 3 - 341 + )

2 ; b - 341 )

2 3b + - 341 )

2 : ! b 3b 4 3 341 + / :

< = % ! !

1 5 # %

# = #% ! + =! -

# # n !- 2 ! " ! / ! !+ - (!" ! n n b $ # = & 12 = ! !+ - (!" ! n k n b1 , b2 , . . . , bk 0 bn−1 = 1 mod n (i = 1, . . . , k) 1! i n−1 1 ;

+ b = 1 mod n - k n b1 , . . . , bk

i 2k ! !+ - n (!" ! 1! 21k #

n ! !+ % 1 − 21k ) " ! 4 ! / # ! ! %+ + - 50 % 1 ! !

n n = pq d = ggT(p−1, q−1) b n bd = 1 mod n n b pq − 1 = (pq − p) + (p − 1) ! "# bn−1 = 1 mod n bd = 1 mod n $ bp−1 = 1 mod q bq−1 = 1 mod p % d = x(p − 1) + y(q − 1) x, y; bd = 1 mod p & '

q ( ) * n = 91 ' b90 = 1 mod 91 b6 = 1 mod 91. + , b6 = 1 mod 7 b6 = 1 mod 13 ! -. b6 = 1 mod 7 b6 = 1 mod 13 / 62 = 36 -. b6 = 1 mod 91 ! / 72 91 0 91 ! ! !% 1 ' 341 ' * * b < 341 (ggT(b, 341) = 1 "# & ! '! -. b10 = 1 mod 341 2 2 11 3 mod 31 3 p q = 2p + 1 4",$ 5 b ' pq / & 6 2 & ! ! d2 ' n = pq + &

-. bd = 1 mod pq ! 7 bd = 1 mod p, bd = 1 mod q ! 8 d -. ! bd = 1 mod pq d2 -. /% 9 ' 0 n = pq ' 50 % n < n

! " #

! " # $ % & ' ( ) ' * n ! + b ' ggT(b, n) = 1' n ,- bn−1 − 1' n n = 2m + 1' n - "- b2m − 1 = (bm − 1)(bm + 1)' () n " - n - & . $ ) n - - / 2' n %' 0 # 1 n ) 2 - &- - bm = 1 - bm = −1 mod n. 1 n " - ! - ,- n ,- bm − 1 ,- bm + 1 ' 1 ) n + b ) bm - 1 bm = −1 mod n' + 2 # + b = 2, n = 341 = 11 · 31' ( 341 " -2 + 2 # 2340 = 1 mod 341' 2170 = 1 mod 341 (m = 170) - - ) 341 " ' - 285 = ±1 mod 341 ' ) 285 = 32 mod 341' 341 ' )

11 ,- 285 + 1 31 ,- 285 − 1 $ - 2170 = 1 mod 341%' # 1 n − 1 = 2a · t t a ≥ 1' b n & $' +' 2% #

(∗)

a−1 t

bn−1 − 1 = (bt − 1) · (bt + 1) · . . . · (b2

+ 1).

n " ) n ,- - & i . '' bt = 1 - b2 ·t = −1 mod n & i 0 a − 1' 3 - , 4 &- 5 - 67' * n 8 " -2 + $--% b n & - & . ,- $9% ' :/ 4 " -2 + b " -2 + b $ -/ ;%' ( - - 2 + 2 " -2 - " -2 $ + 2% ' & #

b = 2 b = . . .!

n = 645 n = 2047 2

2 2047 2! " 2047 = 23 · 89

22046 − 1 = 1023

(2

− 1)(2

1023

+ 1) = (211 − 1) · (211 + 1) = 2047 · 2049 = 0 mod 2047.

# $ % & ' ( $ P2 (x) S2 (x) C(x) 2 %! 2 %! ) *+ < x

,

x 103 104 105 106 107 108 109 1010 25 · 109

P2 (x) 3 22 78 245 750 2057 5597 14884 21853

S2 (x) 0 5 16 46 162 488 1282 3291 4842

C(x) 1 7 16 43 105 255 646 1547 2163

- * ! . . ! - + * / 0 %

% *

+ * .

1 23 4 5 1 21634 2 73 4

)!+ 278 4

15841

+

2 ) 9+ ! 1373653 2 (3, 5)

561 1105 1729 2465 2821 6601 8911 10585 15841 spsp(b) ! ! " b # $ % &' ! ( ≤ 7# ) $ % # * < 15841 ! ! " + +,

# " - $ 65 spsp(8) spsp(18) spsp(14) ) 14 = 8 · 18 mod 65 # ") n = 1069 · 2137 spsp(2) spsp(7) spsp(14)

25 · 109 ! " #$ % &'( ''#) '*+ , , - 2 3 5 . , % / . !(0

25326001 161304001 960946321 1157839381 3215031751 3697278427 5764643587 6770862367 14386156093 15579919981 18459366157 19887974881 21276028621

&.#

&#

& #

/

2251 · 11251 7333 · 21997 11717 · 82013 24061 · 48121 151 · 751 · 28351 30403 · 121609 37963 · 151849 41143 · 164569 397 · 4357 · 8317 88261 · 176521 67933 · 271729 81421 · 244261 103141 · 206281

25 · 109 x 2 x ! " 3 x ! # 5 x ! $ x % & ' ( x x )

(k + 1) · (rk + 1) r k + 1 ! " " " # 2 " $% &

' & ( n psp(2) 2n − 1 spsp(2) )*& n−1 2n−1 − 1 = 0 mod n 2n = 1 mod 2n − 1 22 −1 = 1 mod 2n − 1 + " ," " -' ' + " )' "' + . )'%/ 0%-

& spsp(2) 1 z8 =

108 −7 3

= 33333331

25 · 109

2, 3, 5 7 ! " # $

spsp(11) % n &' (() * + # ' , -' n . #/ 100 $ n ' & %+ 100 n " &' + n $ &'+ 0 # # % 1 # #

1 # ' , .2 3 ) ' " 3 " " % " n 3 $ n 3 3 " n ) "" ' $

" N N − 1 = 2t · d d

" a ad = 1 mod N s ad·2 = −1 mod N s = 0, 1, . . . , t − 1 "" n 2, 3, 5, . . . ( 4"3 3 53 n " $ n " n " 2 20 3 5 20 " n " "" " 3 2, 3, 5 7

" 3 25 · 109 /

2 3 * &2' 4 5 5 # 35 # n + 0 6 n b '- n 1 - ,

n n 25% b 0 < b < n

!"#$% &%' k n

0 n !

n " 1 − ( 14 )k # !

n " #$ k % &' "

( 2400 − l! l = 1, 3, 5, . . . , 591! % 2400 − 593 100 " ) * $ % + " 4−100 < 10−60 # % )% ,%

-" ! $ ! .' ! ( " . + "(! & ( # /' 01 10−60 % ' % ' '

2

& n (

) * + ) ! ,- . +) / ). # b < 2 · (ln n)2 n + ' ! . n < 2! 6 · 1043 '

b < 20000 3 ! n " spsp(b) % 45 " b " %6 7 8 '" ! #$ ' 2 · (ln n)2 2( #$ . b < n 2(&' (

% 9 ' 9 ' ! 4/6 ' + ' 1 )0 $ " ! . 4 & ! - # 6 %

n n

# b < n n

% : n # p > q > 2%

# g p n . b b = g mod p b = 1 mod q % + & 0 %

b n

b = 1 mod n n p q ! " n " # $ %& ' b ( & # $ n ) n = 2c · u + 1 u c > 0

(∗)

c−1

bn − 1 = (bu − 1)(bu + 1)(b2u + 1) · · · (b2

u + 1).

i * q b − 1 bu − 1 (b2 )u + 1 # u = 1, 2, . . . , c − 1 1 + 1 = 2 = 0 mod q (q > 2) + bu − 1 = gu − 1 = 0 mod p , g p ( p − 1 ( u - p − 1 p gu − 1 . ) / 0 spsp(b) / ) 1 )2 n = pa (a > 1) n " ! 3 4( & b n " & b ($ ( g < n gj = 1 mod p2 # j < (p − 1)p ! 2 g p2 5 5 "" Z/p2 Z &" 2 )# b = g 2 gn−1 = 1 mod n $ gn−1 = 1 mod p2 p(p − 1) n − 1 = pa − 1 = (pa−1 + pa−2 + . . . + p + 1)(p − 1) 6 2

7 $ / # # $ 3 4 3 4(2 n ( ! n 8 > 1 n 3 4 %3 4( 9 ' ! n 9 n 3 # p n ( p − 1 n − 1 1 & n p > q & ) 5: g /

# g % 9 ; ' p x 1 p − 1 x2 = g mod p / # & $ # +2 n = 91 25%

11 10761055201 6

! " # $ spsp(2)

%

# & 561 '

() *+

2p − 1 p

! "# $ %&&'( )

! M32582657 = 232582657 − 1 $* *

+ ,

-./+( ./+ ! + # 0 * )

./+ + . 1 1)

M32582657 "# )

1 # )

!

2 Mq q = 39051 · 26001 − 1 q "#! $ 33'( )

+ 2. 2 $4 56 + 6 ( 1 7 8 ! ./+ 1 * 1 9 ) , : 8 ; ; Z[φ] < n

n

r(n) := φ2 + (1 − φ)2

$= (

1 <

r(1) = φ2 + (1 − φ)2 = φ2+1 − 2φ + φ2 = 2(φ2 − φ − 1) + 3 = 3. = m

m

m+1

(φ2 + (1 − φ)2 )2 = φ2

m

m+1

+ 2(φ(1 − φ))2 + (1 − φ)2

= r(m + 1) + 2.

r(m + 1) = r(m)2 − 2

7 < 3 7 47 2207 4870847 23725150497407

r(m) r(m) = 0 mod p r(m + 1) = −2 mod p r(m + 2) = 2 mod p n φ2 n

n+1

φ2 · r(n) = φ2

n

n+1

+ (−1)2 = φ2

!"

+1

# $ % & ' (

p

r(p − 1)

M

4n + 3

M = 2p − 1

% M ) %

M = 2p − 1 = 24n+3 − 1 = 8 · 16n − 1 = 8 − 1 mod 10 = 7 mod 10 5 M & *

+ $ , R = (Z/M Z)[φ] - , . /0 φM = (1 − φ)

φM +1 = φ(1 − φ) = −1 mod M. . p − 1 1 n ! $ p−1

φ2

· r(p − 1) = φM +1 + 1

. 0 R φ R % r(p − 1) M $ 2 2& . + 3 $ $ '

!

q

5

5 2n+1

r(n)

q q −1

q

q+1

2n+2

+ $ ! Z[φ] 5 4

, q $ Z/qZ (5 a

x2 − x − 1 = 0 $ 6 , ρ : Z[φ] → Z/qZ ρ(φ) = a. - Z/qZ r(n) q $ n

n+1

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n+1

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Z/qZ 0 1

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2 3 (r(n)|n ∈ N) 4 5 - - 6 - ! (

M = 2p − 1

r(p − 1)

M

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