. -2 "tex" 1 Carl Friendrich Gauss (1777-1855) Disquistiones
Arithmeticae, 24 - . , , Pierre de Fermat(1601-1665) - . , modm, m,
m ( ) mod m. m ( ), , modm mod m. . 1 a, b, c, d .:1. a b mod m b a
mod m a b 0 mod m.2. a b mod m b c mod m a c mod m.3. a b mod m c d
mod m (a+ c) (b + d) mod m.4. a b mod m c d mod m ac bd mod m.5. a
b mod m d m, d > 0 a b mod d.6. a b mod m, ac bc mod m, c >
0. 2 f (x) Z[x]. a b mod m, :f (a) f (b) mod m.34 1. 3 , x, y Z m1,
m2, . . . , mr, m N.1. x y mod m x y modm(,m).2. x y mod m (, m) =
1, x y mod m.3. x y mod mi, i = 1, 2, . . . , r x y mod [m1, m2, .
. . , mr] , (x, y) = ...x, y [x1, . . . , xr] = ... x1, . . . , xr.
2 Fermat, Euler Zn= 0, 1, . . . , n 1 modm. Zn :+ : + = +, : = (Zn,
+, ) . , (Zn, +) (Zn, +) = (), (, n) = 1. , (Zp, +, ) p. Zn= Zn :
(, n) = 1. (Zn, ) () . , Zn , Z : + n=1 = 1 (, n) = 1, 1= . 1 1.
Z5= 0, 1, 2, 3, 4 Z5= 1, 2, 3, 4. 2 : 2, 22= 4, 23= 3, 24= 1 ,
Z5=_2_. ,Z5=_3_, 33= 2 , _4_ = 4, 1 Z5.2. Z8= 1, 3, 5, 7 , 32= 1,
52= 1, 72= 1, 4. 4(Fermat)p Z,p. p1 1 mod p., Z/0 pmod p. 56 2.
FERMAT, EULER, , . (Zp, ), Zp = p1. Zp, p1= 1. , p p1 1 mod p. p1 1
mod p p mod p. p . Z/0, pmod p, p .
5(Euler) n> 1 Z (, n)= 1. (n) 1 mod n, Euler, (n) =Zn..
(n)=Zn, Zn, (n)= 1, Z, (, n) = 1, (n) 1 mod n. 6( Wilson)(i) p (p
1)! (1) mod p(ii) n > 1 , (n 1)! (1) mod n, n ..(i) (Zp, +, ) p1
. Zp, 2 x2=1 Zp, x2 1=0Zp. 1 1=p 1. , Zp 1 p 1 , aa1. , 12 p 1=p 1,
(p 1)! (p 1) mod p (p 1)! (1) mod p.(ii) n > 1 , (n 1)! (1) mod
n. n , n=n1n2 1 1 .:(i) ZnmZn Zm, (ii) ZnmZn Zm.910 3. EULER.(i) f
: ZnmZm Zm, a mod nm (a modn, b mod m). f -. f 0 mod nm. ,f (a mod
nm) =(0 mod n, 0 mod m) a 0 mod n b 0 mod m. (n, m)= 1, a 0 mod nm.
, f . (b mod n, c mod m) Zn Zm. , a, a b mod n , a c mod m a modnm.
f (a mod nm) = (b mod n, c mod m). f .(ii) f , Znm f Zn Zm. Znm =Zn
Zm. 9(i) n > 1, m > 1 . (nm) = (n)(m).(ii) n > 1 n=
si=1piei , pi, 1 i s , n. (n) =
si=1_piei piei1_ = n
si=1_1 1pi_..(i) 8(ii) n>1, m>1 , , (nm) = (n)(m).(ii) (i)
(pe) = pe pe1. 10 n > 1 . n=_dn (d), Euler..n>1, 1n, 2n, 3n,
. . . ,n1n,nn, n. ad (a, d) = 1. n d (d), Euler. n=_(d).111. m N m1
m m1 > 1. a mod m1 nm1 : a, (a+ m1) mod m, . . . ,[a +_mm1 1_m1]
mod m. (a+ km1) mod m, 0 kmm1 1 a mod m1. ., a+ m1 (a+ m1) mod m 0
, mm1 1 ( )m1 0 mod m modmm1., Z=_m11a=0a mod m1 Z=_m1=0 mod m a
mod m1 . a mod m1 (a= km1) mod m, 0 k mm1 1. , x a mod m1 x= a + m1
, Z x= a +_mm1+ _m1,=_mm1+ _, 0 mm1 1, x =a+ m+ m1, 0 mm1 1, x (a +
m1) mod m, 0 mm1 1. 1 2 + 9Z = (2 + 18Z) _(11 + 18Z), m= 18, m1=
9.2. n, m N/0 (n, m) = 1. , , a1, . . . , an1 (- {1, . . . , m1)
modn (- modm) . aim +jn, 1 i n 1, 1 j m 1 modnm. aim+ jn, 1 i n 1,
1 j m 1 nm, modnm. aim+jn (am+n) mod nm nm [(ai a)m+ (j )n] j mod m
ai a mod n. 1 : m N/0 0 a m1 - modm, sa+ , 0 a m 1, (s, m)= 1 Z,
modm.12 3. EULER 2 - .3. n1, n2, . . . , ns . Zn1 Zn2 Zns Zn1n2ns
.: : s 8. : Z Zn1 Zn2 Zns,a (a mod n1, a mod n2, . . . , a mod
ns)4. (n+ 1)2x 1 mod (2n + 1)3, n Z._(n + 1)2, (2n + 1)3_ = 1. , p
p (n + 1)2 p (n +1). , p (2n +1)3 p (2n +1) = 2(n +1) 1 p 1,. .
:(2n + 1)3= (n + 1)2(8n 4) + 6n + 536(n + 1)2= (6n + 5)(6n + 7) + 1
1 = 36(n + 1)2 (6n + 5)(6n + 7) = 36(n + 1)2 [(2n + 1)3 (n + 1)2(8n
4)] = (n + 1)2[8(6n2 4n + 1)] (2n + 1)3(6n + 7). 8(6n2 4n + 1) mod
(2n + 1)3 .5. aix i mod mi, 1 i s, m1, m2, . . . , ms , (a1, m1) =
1, i, ai Z. , aix i mod mi, 1 i s, , x ci mod mi, 1 i s. , ,
modm1m2 ms.6. aix i mod mi, 1 i s, m1, m2, . . . , ms ai, i Z.aixi
mod mi, 1 i s di:=(ai, mi) i, 1 is . di, 1 i s , : ci, ci+midi, ci+
2midi, . . . , ci+ (di 1)midimodmi, ci aidix
idimodmidi, 1 i s. d1, . . . , ds, 13 modm1 ms, .7. 3749 7.
Fermat, 3771 1 mod 7, 376 1 mod 7. 49= 68+ 1, 3749=(376)8 37 37 mod
7=(57+2) mod 7 = 2 mod 7.8. 347 23. Fermat 322 1 mod 23.47 = 222 +
3, 347 33mod 23 = 4 mod 23.9. 2217+ 1 19. Fermat 218 1 mod 19.
217mod 18. (2, 9) = 1, 26 1 mod 9 272 mod 18 217=227+3=(27)2 2322
23mod 18 =25mod 18 = 32 mod 18 = 14 mod 18. 217 14 mod 18 217=
14+k18, k Z , 2217= 214+k18 214mod 19. 218= 1 mod 19 214 24 1 mod
19, 214 24 Z19., 19 = 16+3, 16 = 3 5+1 11 = 163 5 = 165(1916) ==
519 +616, 6 mod 19 16 mod 19. 2217 214mod 19 6 mod 19 2217+ 1 7 mod
19.10. p , p 3. (p 2)! mod p. Wilson, (p 1)! (1) mod p. , (p 1) (1)
mod p. , (p 2)!(p 1) 1 mod p (p 2)! 1 mod p.11. 34! mod 37Wilson,
36! (1) mod 37. , 10, 35! 1 mod 37. , 34!35 1 mod 37 34! (35)1mod
37. , 37 = 35+2, 35 = 217+1 1 = 1737+1835, 1835 1 mod 37, 34! 18
mod 37.14 3. EULER12. 49! mod 53.(53 2)! 1 mod 53 51! 1 mod 53
49!5051 1 mod 53.5051 =2550Z53. 2550 =4853 + 6, 53= 86 + 5, 6= 5 +
1 1= 92550 43353. ,49! 9 mod 53.13. n Z, 383838 (n37
n).383838=237131937 383838 . 37 n 37 (n37 n). 37n n36 1 mod 37 37
(n36 1) 37 (n37 n). , 2, 3, 7, 13, 19 (n37 n), 383838 (n37 n). 2. .
-. :1. 112. 12, .82:12.1, 12.2, 12.4, 12.53. 14, .91:14.1, 14.2,
14.4, 14.5, 14.6, 14.84. 15, .105:15.4, 15.5 4 (Zp, ) (Zp, ) -. ,
.G a G. a, ord(a) s0 :as=e, e G. (a) , G, a. - : 1 G= (a) n <
.(i) ord(a) = (a),(ii) a= e n (iii) a= a mod n(iv) ord(at)
=n(n,t)(v) G= (at) (n, t) = 1(vi) G (vii) m n Gm, H= (anm). 2 G ,a
G p . ap= e ap1 e, ord(a) = p..m=ord(a), ap=e (ii) 1, 1516 4. (ZP,
) m p. m=pt, 0 t . t 2 . Legendre :(a/p) =___1, pa a modp0, p a1,
pa a modp x2=a mod p (a, p)= 1 Zp, x2 a Zp[x] . , (a/p) =1, p a
(a/p) = 1. , (a/p) + 1. Legendre 14 p a, Z.:(i) (a/p) ap12mod p(ii)
(1/p) = (1)p12(iii) (a/p) = (a/p)(/p)(iv) a mod p (a/p) = (/p)(v)
(a2/p) = 1 (1/p) = 1..(i) Euler ( 13).(ii) (1/p) 1. (1)p12
Euler.(iii) p ap , (a/p) =0 (/p) =0. p a p , Euler (a/p)(/p)=
ap12
p12= (a)p12=(a/p) mod p p > 2 (a/p) 1, (a/p)(/p) = (a/p).(iv)
Legendre.(v) (iii) . 4 x2 63 mod 11.63 8 mod 11, (63/11) = (8/11).
(8/11) = (23/11) = (2/11) (22/11) =(2/11)1=(2/11). , 32 Z112,
(2/11) = 1, .24 5. - LEGENDRE 4 1. . - . ,17:17.3 , 17.4 , 17.52. 3
modp p= 7, 13;3. (a/p) a= 1, 2, 2, 3 p= 11, 13, 17.4. ;(i) x2 2 mod
61(ii) x2 2 mod 59(iii) x2 2 mod 61(iv) x2 2 mod 59: , . 15( Gauss
Gauss) p pa. a, 2a, 3a, . . . ,p12a modp. n p2, (a/p) = (1)n.. r1,
r2, . . . , rn modp a, 2a, . . . ,p12a - p2 s1, s2, . . . , sk .
r1, r2, . . . , rn, s1, s2, . . . , sk . , n+ k=p12. ,ri>p2 0 1
. N= 5(n!)21. N+1 = 5(n!)2, N 1 mod 5. p N, p1 mod 5.-, N 1 mod 5
N=
si=1paii, n=
si=1(1+5i)ai, ai, i, 1 i s., N 1 mod 5, N N 1 mod 5, . p N p1
mod 5. pp>n. p N N 0 mod p N+ 1 1 mod p _N+1p_=_1p_= 1
1=_N+1p_=_5(n!)2p_=_5p_ _(n!)2p_=_5p_.,_5p_ _p5_=(1)p12512=1.
_5p_=_p5_, 1=_5p_, 1=_5p_=_p5_, p mod5. mod 5 1 4, Z52= 1, 4.p 4
mod 5, p1 mod 5. , , p < n.,p n!, p 5(n!)2. , p NN= 5(n!)2 1, p
1, . ,, n> 1, p>n, p 4 mod 5. 5m 1.299. _8745231_. 874 =
(1)21923._8745231_ =_15231_ _25231_ _195231_ _235231_ ==
(1)523112(1)5231218(1)1912523112_523119_ (1)2312_523123_ ==
_523119_ _523123_.,_523119_ =_ 619_ _523123_ =_1023_._ 619_ =_2319_
=_ 219_ _ 319_ = (1)19218(1)3121912
_193_ =_193_ =_13_ _1023_ =_1023_ _1023_ ==
(1)23218(1)5122312
_235_ =_235_ =_35_ = (1)312512_53__53_ =_23_ = (1)3218. .10. -3
modp, p > 3 p= 6m+ 1, m N._3p_= 1 _1p_ _3p_= 1 (1)p12_3p_= 1
_3p_=(1)p12. , _3p_ _p3_=(1)p12312. _p3_=(1)p12(1)p12_p3_= 1. pp 1
mod 3p 2 mod 3. p 2 mod 3, _p3_=_23_= 23218= 1, , _p3_= 1.p 1 mod 3
3 (p 1). , 2 (p 1), p ., 6 (p 1) , p= 6m+ 1., p= 6m +1 m N, (3/p)
=_1p_ _3p_ =(1)p12_p3_ (1)p12312=_p3_=_6m+13_=_13_=1, -3 modp p 6m+
1.- 8. , , - . 18(Dirichlet)a,N, (a, ) = 1. , an+, n N.11. p > 2
3 ;_3p_ =_p3_ (1)p12,_p3_ =____13_ = 1, p 1 mod 3_23_ = 1, p 2 mod
3.30 6. ,(1)p12=___1, p 1 mod 41, p 3 mod 4._3p_ = 1 ___p 1 mod 3,
p 1 mod 4p 2 mod 3, p 3 mod 4.,_3p_ = 1 p 1 mod 12 p 11 mod 12.
5ap> 2, pa. (a/p)= (1)m, m=_p12i=1_ia/p_ [x] x.. , ia =ip+ i, 0
i 1. , f (x) Zm[x]. f (x) = anxn+an1xn1+. . . +a1x +a0 Zm[x] n an 0
Zm. , f (x) 0 mod m n. a Z, f (a) 0 mod m. a f (x) mod m.
19(Lagrange) f (x) Z[x]n. f (x) 0 mod p, p , n ..Zp , f (x) Zp[x]
n, n. , f (x) 0 mod p n . 7 1. xpx 0 mod p p , Fermat a Zp, ap a a
mod p.2. x2 1 0 mod 8 1, 1, 3, 3Z8, . Lagrange .3. 8x3+2x2+1 0 mod
5 3, 8x3+3x2+x+ 1 0 mod 4 2.3334 7. 20 d m, d> 0 a f (x) 0 mod
m, a f (x) 0 mod d..m f (a) d f (a) f (a) 0 mod d. 21f (x).
-mN(m)-f (x) 0 mod m. m=m1m2, (m1, m2) = 1,N(m) =N(m1)N(m2).,
m=
ti=1peii, N(m) =
ti=1N _peii_.. a Zm f (a) 0 mod m, . m= m1m2, (m1, m2)= 1, f
(a)= 0 mod m1m2 mi f (a), i = 1, 2 f (a) 0 mod mi, i = 1, 2. a1
Zm1, a a1 mod m1 a2 Zm2, a a2 mod m2. , 2, f (a1) 0 mod m1 f (a2) 0
mod m2. , a f (x) 0 mod m, (a1, a2), ai f (ai) 0 mod mi, i = 1, 2.
, a1, a2 f (x) 0 mod m f (x) 0 mod m1 f (x) 0 mod m2. , a mod m 1
mod m1 2 mod m2, a1 1 mod m1 a2
2 mod m2. (m1, m2) = 1. . . ai f (x) ai mod mi. , a Zma a1 mod
m1 a a2 mod m2. f (a) 0 mod m1 f (a) 0 mod m2 f (a) 0 mod m1m2,
(m1, m2)= 1, f (a) 0 mod m. N(m) = N(m1)N(m2). 2 f (x) 0 mod m,
N(m) = 0. 8 1. f (x) = x2+x +7 0 mod 15. x2+x+7 0 mod 3 x2+ x+ 7 0
mod 5. 0,1,2 x2+ x+ 7 0 mod 3 , x 1 mod 3. ,35x2+ x+ 7 0 mod 5 , x=
0, 1, 2, 3, 4( 0,1, 2)., 2. x2+x +7 0 mod 21. x2+x +7 0 mod 3 , x 1
mod 3, .x2+x+7 0 mod 7 , x 0 mod 7 x 1 mod 7. 12= 2 . (1, 0), ___x
1 mod 3x 0 mod 7 (1, 1) (1, 6) ___x 1 mod 3x 6 mod 7.,
1M1213+0M2217 , M1213 1 mod 3 M1= 1 M2217 1 mod 7 M2= 5. , 117 +
053=7 mod 21. 117+653 = 97 mod 21, 13 mod 21. 5 1. f (x) =x2+ x+ 7.
f (x) 0 mod 189, 189= 33 7 f (x) 0 mod 27 4 13.2. x3+ 4x+ 8 0 mod
15. f (x) 0 mod pi, f (x) Z[x], p i 2. 22( Hensel) f (x) Z[x]. a
mod pi f (x) 0 mod pi p f(a)0 mod p, f(x) f (x), t mod p f (a+ tpi)
0 mod pi+1, i 1. t tf(a) f (a)pimod p.36 7. . f (x)= k0 + k1x+ . .
. + knxn Z[x]. f(x) f (x), f(x) =k1+ 2k2x+ . . . + nknxn1. - a mod
pi f (x) 0 mod pi, f (a) 0 mod pi. t mod p (a+ tpi) mod pi+1 f (x)
0 mod pi+1. Taylorf (a+ tpi) = f (a) + tpif(a) +12!t2p2if(a) + . .
. +1n!tnpnif(n)(a), n f (x). f (a+tpi) (f (a) + tpif(a))
modpi+1(1). , pi psis 1. 1s!f(s)(a) 2 s n. r f (x) krxr, f(s)(a)
krr(r 1)(r 2)(r s +1)ars, 0 r n. s! r(r 1)(r s +1)(
_rs_=r(r1)(rs+1)s!Z). , 0 r s, _rs_ s! (1). , , a+ tpi f (x) 0 mod
pi+1, f (a +tpi) 0 mod pi+1, (1) :f (a)+ tpif(a) 0 mod pi+1f(a)t f
(a)pimod p. f (a)pi, f (a) 0 mod pi . , , f(a) 0 mod p, t f
(a)f(a)pimod p, f(a) f(a) mod p, f(a)f(a) 1 mod p. , , f(a)0 mod p,
a +tpi f (x) 0 mod pi+1, t (1). , , (1) a+ tpi [a f (a)f(a)] mod
pi+1(*). 5x2+ x+ 47 0 mod 73. f (x) =x2+x+ 47. Z7a1 mod 7 5 mod 7f
(x) 0 mod 7. , f(x) =2x+ 1 f(1) =30 mod 7, f(5) =110 mod 7. -Hensel
t a+ 7t f (x) 0 mod 72t + 7tf (x) 0 mod 72.a 1 mod 7, f(1) 3 mod
7f(1) 5 mod 7, 35= 15 1 mod 7., (*), 1+7t= 1f (1)f(1) 1+7t= 149375
(1245) mod 72 1+7t 244 mod 49 1+7t 1 mod 49.a2 1 mod 49 a2 f (x) 0
mod 72., 5 mod 7, f(5) 11 mod 7 f(5) 4 mod 7f(5) 2 mod 7, 42= 8 1
mod 7. , (*), 5 + 7t=5 f (5)f(5) 5 + 7t= 5 772 2 mod 49 5 + 7t 47
mod 49. 2 47 mod 49., a2 1 mod 72 2 47 mod 72a3 mod 73 3 mod 73 x2+
x+ 47 mod 73. a2 1 mod 72, f(1) = 3, f(1) 5 mod 7 a3= a2f (a2)f(1)
a3= 1 495 mod 73 a3= 245 mod 343 a3 99 mod 343 a3 99 mod 73. 2 47
mod 72, f(47)= 95 4 mod 7 f(47) 2 mod 7
3= 2f (2)f(47) 3= 472303 2 mod 73 3= 4551 mod 343
3 243 mod 343 3 243 mod 73. , - : 99 mod 73, 243 mod 73. 3
Hensel (a + tpi)modpi+1, f (a) 0 mod pi f(a)0 mod p. f(a) 0 mod p,
Taylor f (a +tpi) f (a) mod pi+1 t. , f (a) 0 mod pi+1 f (a+ tpi) 0
mod pi+10 t p, p , a+tpi, 0 t p., ,f (a)0 mod pi+1, a +tpi ., a mod
pi f (x) 0 mod pi+1. 6f (x) =x2+ x+ 7 0 mod 81, x2+x +7 0 mod 34.
x2+ x+ 7 0 mod 3 a 1 mod 3 f (1) =9.f(x)= 2x+ 1 f(1)= 3 0 mod 3. ,
f (x) 0 mod 32 1 mod 32, 1+3 = 4 mod 32 1 + 23 = 7 mod 32.a1 mod
32, f (1) =90 mod 33. f (x) 0 mod 33 a 1 mod 32. 4 mod 32, f (4) =
27 0 mod 33, 4 mod 32f (x) 0 mod 33. f(4) = 24 + 1= 9 0 mod 3, ,, f
(x) 0 mod 33 4, 4 + 32= 13 4 + 232= 22., 4 mod 32 4 mod 33, 13 mod
3322 mod 33, f (x) 0 mod 33.38 7. 1 4 mod 33, f (4) = 270 mod 34. 2
13 mod 33, f (13) = 189 27 mod 34 f (13)0 mod 34. 3 22 mod 33, f
(22) = 513 27 mod 34 f (22)0 mod 34. x2+ x+ 7 mod81.f (x) 0 mod p,
f (x) p -. Lagrange n, n =deg f (x). x2+ x+ 7 0 mod 33 3 , ,
Langrange f (x) Zn, n . f (x) Z[x] f (x) mod p n p n p. - f (x)
Z[x] xp x Z[x]. q(x), r(x) :f (x) = q(x)(xpx)+r(x), 0 deg r(x) p. x
Z xp x mod p, Fermat, f (x) 0 mod p r(x) 0 mod p. , r(x) = 0 r(x)
p, r(x) 0 mod p. , r(x) - p 1. r(x)=a0+ a1x+ . . . + asxsZ[x], s p
1 pas, r(x) 0 mod p asa0 + . . . + asas1xs1+ xs 0 mod p, as as Zp.
, asas 1 mod p pas. , r(x) 0 mod p asr(x) 0 mod p. f (x) 0 mod p
asr(x) 0 mod p. . 23 f (x) 0 mod p n n p. f (x) 0 mod p g(x) p1,
g(x) 0 mod p . 6 1. x3 3 0 mod 52. 2x3 39 + 9 0 mod 73. 7x4+ 19x+
25 0 mod 27394. 23 :(i) x11+ x8+ 5 0 mod 7(ii) x20+ x13+ x7+ x 2
mod 5(iii) x15 x10+ 4x 3 0 mod 7.40 7. 8 f : N/0 C . :(n) = n(n) =
nk(n) = k nw(n) = n(n) = n ., (8) = 4, (12) = 6, (8) = 1+2+4+8 =
15, 2(8) = 1+22+42+82=85, w(8) = 1(8) = 3. :(n) =_dn 1, (n) =_dn d,
k(n) =_dn dk, w(n) =_pn 1, (n) =_pan a, pa n a p n. f f (mn)=f (m)f
(n), (m, n) = 1 . - f (mn)= f (m)f (n) m, n . 1 1. f f (n) =f (n1)
f (n) = f (1)f (n) f (1) = 1, C nf (n) 0. , n =
ti=1peii, f (n)=
ti=1f (peii). , f f (peii) , 1 i t.2. f n =
ti=1peii. f (n)=
ti=1f (pi)ei, f f (p), p4142 8. . 9 1. Euler .2. . . 6n=
p pa(p) n N 0, n , pa(p) n. (n) =
pn (a(p) + 1)..d n d=
p p(p) 0 (p) a(p), pn., (p) a(p) + 1., n pn (a(p) + 1).3. I : N
0 C, I(n) =___1, n= 10, n > 1 .4. J : N0 C, n 1 .5. : N 0 C,
:(n) =___0, 1 n(1)r, n r M obius. (1) = (1)0= 1, (2) = 1, (4) = 0,
(6) = 1. Dirichlet( convolution, ) :f g(n) =_dn f (d)g(nd), f g n
N0. Dirichlet 43 , f g . d g(n)=_n=d1d2 f (d1)g(d2), (d1, d2),
d1d2= n. Dirichlet , f g(n) = g f (n)., , [(f g) h](n)=[f (gh)](n)
f, g, h. [(f g) h](n) =_n=d1d2d3 f (d1)g(d2)h(d3) = f (gh)(n), (d1,
d2, d3) n= d1d2d3. I f (n) =_dn I(d)f (nd) =I(1)f (n) =f (n), I f
=f =f I. . 7 Dirichlet - I, : I(1)= 1 I(n)= 0 n > 1. - . 24 f ,
F(n) =_dn f (d) .. F(m1m2)= F(m1)F(m2), (m1, m2)= 1. m=m1m2. dm
(d1, d2), d1 m1 d2 m2. d m. d1=(d, m1) d2=(d, m2). d=d1d2, d1 m1 d2
m2. ,, d (d1, d2)., (d1, d2) d1 m1 d2 m2, d=d1d2 m= m1m2 d1= (d,
m1) d2= (d, m2). (d1, d2) d= d1d2. . d : d m1m2 =(d1, d2) : d1 m1,
d2 m2. , (d1, d2) = 1, F(m1m2) =_dm1m2 f (d) =_d1m1_d2m2 f (d1d2)
=_d1m1_d2m2 f (d1)f (d2) =__d1m1 f (d1)_ __d2m2 f (d2)_ =
F(m1)F(m2).44 8. 10 1. (n) =_dn 1. (n) =_dn J(d), J(n) =1, n1 J . ,
24 . n =
ti=1peii, (n)=
ti=1d(peii)=
ti=1 (ei+ 1) (. 2).2. 24 (n). (n) =_dn d=_dn f (d), f (n) =n. f
, ., n=
ti=1p1iei, (n) =
ti=1(peii) =
ti=1(1 + pi+. . . + deii) =
ti=1pei +1i1pi1. : 25 n=
ti=1peii, (n) =
pei +1i1pi1.3. _dn (n) = n, n1 Euler. F(n) =_dn (d). , F .
n=
ti=1peii, F(pe).F(pe) =_dpe (d) =_es=0(ps) = 1 +_es=1ps ps1=pe(.
3. n> 0 , (n) = 2n. . 8 n> 1 s> 1 . ns 1 n > 2 s .. s=
tu, t > 1 u > 1. ntu 1= (nt 1)(nt(u1)+ . . . +nt+ 1), ns 1 ,
n > 2 s . Mp= 2p 1, p , - Mersenne( Father Martin Mersenne ,
1588-1648). Mersenne, -45 . Mersenne . 9 n 2p1(2p 1), p 2p 1 . 4 4
.. . Euler 18 ... n= 2p1(2p 1) p 2p 1 ( 8, 2p 1 , p ). n : 1, 2, .
. . , 2p1, 2p1, 2(2p 1), . . . , 2p1(2p 1). , 1 + 2 + . . . + 2p1=
2p 1 1 + 2 +. . . + 2p1+ (2p 1) + 2(2p 1) + . . . + 2p1(2p 1) ==
(2p 1) + (2p 1) + 2(2p 1) + . . . + 2p1(2p 1) =2p 1 + (2p 1)(2p 1)
= 2p(2p 1) = 2n. (n) = 2n, n ., n . n=2k1m,k2, m (n) =2n. , 25. (n)
=(2k1)(m). ,(n) =2n, (2k 1)(m) =2km (2k 1) m, m=(2k 1)m1., (m) =
2km1. m m1 m., m1= 1 , m= 2k1 , k , 8, , n= 2k1(2k 1). 5 . <
106: 6, 28, 496 8128.p= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257 Mp
-, p < 257. . 24 : f , f J . 46 8. , J f (n) =_dn f (d) -f . ,
(. 1) (n) =_dn J(d) =J J(n), = J J. M obius. 26 f n=
ti=1peii _dn (d)f (d) = (1 f (p1))(1 f (pt)).( t= 0, n= 1, 1)..
(d) d 1, 2, . . . , t. , .. d =q1 qs, q1, . . . , qs p1, . . . ,
pt, (d)= (1)sf (d)=f (q1)f (qs),(d)f (d) =(1)sf (q1)f (qs). , . 11
1. f (n) =1n, n N0, n=
ti=1peii_dn (d)1d= (1 1p1 )(1 1pt).2. f = J., 26 :J(n) =_dn
(d)J(nd) =_dn (d) =_dn (d)I(d) =
ti=1 (1 I(pi)) =
ti=1(1 1) =___1, n= 10, n > 1= I(n). J= I = J . 27(M obius) f
F . :F= J f f = F.. F= Jf F= (Jf ) F= (J)f F= I f F= f , I f =f =f
I J=I =J . , f = F J f = (J ) F J f =I F=F, I f = f = f I J= I = J
., , 24. 28F F(n) =_dn f (d), f , f .47.F(n) =_dn f (d) F =f J(. J
f (n)=_dn f (d). , M obius F = f J F = f J= I = J . , F( ) , f . 7
1. Dirichlet - .2. :(300), (250), s(ee 52 7).3. J f (n) =_dn f (d),
f - J(n) = 1 n 1.4. _ti=1peii_=
ti=1piei_1 1p1_
_1 1pt_, M obius. (: F(n)= n f (n) = (n).)5. _dn (d)_nd_=1 _dn
(d) (d) =(1)t, n=
ti=1peii. (: =J J J= I = J .)6. _dn (d)_nd_= n _dn (d) (d)=
(1)tp1 pt, n=
ti=1peii.7. (n) = n_dn (d)1d.48 8. 9 , 2 . 4 a , - f (x) Q[x], f
(a) = 0. 122 , x2 2. i x2+ 1. a=2+335 a 2= 335 (a 2)3= 275 a3 6a2+
12a 8= 135 a3 6a2+ 12a 143 = 0. 2 + 335 , x3 6x2+ 12x 143 Q[x]. ,
-. e, , . , -.(;) 29a. -() p(x) a. a p(x). : F: Q[x] C, f (x) f (a)
-, , (x), Q[x] . 4950 9. Q[x]a (x). (x) . , (x) =1(x)2(x), 0 = (a)
= 1(a)2(a) 1(a) = 0 2(a) = 0, C . (x) 1(x) (x) 2(x). , 1(x) 2(x) Q,
Q[x]. (x) . (x) 1q Q x, 1q(x) - a. p(x) 1q(x). g(x) a, p(x) g(x).
p(x) =g(x) p(x) g(x) g(x) . 5 a Q[x] a a Q IrrQ(a)(x). 7 x27 =
IrrQ(7), x2+1 = IrrQ(i)(x), x2+x+1 = IrrQ_1+i32_ =IrrQ_1i32_ (x),
x2+ 3 = IrrQ(i3)(x). 8-:1+372, 1 +2 +3, 1 + 3i 2 Q[a] 29. Q[a] (C)
Q[a] = f (a) : f (x) Q[x]. Q(a) - Q[a], Q(a)= f (a)g(a):f (x), g(x)
Q[x], g(x) 0. 29 , Q[x]/ (p(x))Q[a]., Q[x]/ (p(x)) (p(x)) , p(x) =
IrrQ(a)(x). Q[a] Q[a]=Q(a), Q[a] , Q a. Q(a) Q a. a , Q[x]Q[a]
Q(x)Q(a), Q(a) - Q, Q(x) Q[x].51 30a n IrrQ(a)(x). 1, a, . . . ,
an1QQ(a), Q(a) = qo+ q1a+ . . . + qn1an1: qi Q, 0 i n 1. : 1, a, .
. . , an1 -, Q[x] a n 1. , degIrrQ(a)(x) =n. 1, a, . . . , an1 . ,
Q(a)=Q[a] f (a) f (x) Q[x]. , f (x) = (x)IrrQ(a)(x) +(x), (x), (x)
Q[x], deg(x) < n,f (a) =u(a), f (a) - 1, a, . . . , an1, Q(a) 1,
a, . . . , an1, Q . 13 1. Q(2) = q0+q12 : q0, q1 Q, IrrQ(2)(x) =x2
2.2. , Q(i) = q0 + q1i : q0, q1 Q.3. IrrQ()(x) =x2+ x+ 1, 3. Q() =
q0 + q1 : q0, q1 Q.4. IrrQ(37)(x) = x37, Q(37) = q0+q137 +q2372:
q0, q1, q2 Q.x3 737,37,372, 1, , 2 x3 1. IrrQ(37)(x)=
IrrQ(37)(x)=IrrQ(372)(x) = x3 7.5. q Q, IrrQ(x) = x q. q Q. Q := a
C : a . Q Q C. 6a -, IrrQ(a)(x) Z[x]. Z = a Q : a . 14 1. 2, i,37,
3- , .52 9. 2. 3+i154 . ,a=3+i154 2a2+ 3a+ 3= 0,2x2+3x +3 Q[x] a.,
IrrQ(a)(x) = x2+32x +32
Z[x].3. a Z Z, x a Z[x]. 31(i) Q C.(ii)a Zf (x) f (a) = 0.(iii)
a Q .(iv) Z Q..(i) a, Q. , f (x), g(x) Q[x]f (a) =0g() = 0. f (x)
=k ni=1 (x ai) g(x) =
mj=1 (x j) f (x) g(x) C[x],
n,mi=1,j=1 (x (ai+ j)),
n,mi=1,j=1 (x (ai
j)) a+ a ai, i i n j, 1 j m, Q, Q[x]. - C Q . a Q, a Q 0a, a1 Q,
Q C. f (x) =q0+q1x+. . .+qnxn Q[x] f (a) = 0, a q0 q1x+ q2x2+ . . .
+ (1)nqnxn. a Q. , a 0, q0 + q1a+ . . . + qnan= 0 an_qn+ qn1 1a+ .
. . + q01an_= 0, a1 qn +qn1x +. . . +q0xn Q[x]. Q C.(ii) a Z, . , f
(x) Z[x] a. IrrQ(a)(x), - g(x) Q[x], f (x)=IrrQ(a)(x)g(x). g(x)
Gauss IrrQ(a)(x), g(x) Z[x]. a Z.(iii) a Q IrrQ(a)(x) =q0+q1x+. .
.+qnxn. m- 53IrrQ(a)(x), 0=mnIrrQ(a)(a) =mnIrrQ(a)_qmm_. am=. 0 =
mn_q0 + q1
m+ . . . + qn1
n1mn1+
nmn_ ==q0mn+ q1mn1+ . . .+ qn1mn1+ n. , q0mn+ q1mn1x+ . . . +
qn1mxn1+ xn Z[x]. Z a=
m, .(iv) , (i), a, Z, a a Z. Z - Q . (iii) Q Z. 10 Z Q = Z.. a Z
x a Z[x] a Z. Z Z Q. a Q Z x a Q[x] a Z, IrrQ(a)(x)= x a Z[x] x a
Z[x] a Z, Q Z Z. , Z Q = Z. Q Q QQ . 11_Q : Q_ = ..n p . xn p Q[x]
_Q_np_ : Q_= n. Q Q_np_ Q, . n Q n Q, _Q : Q_ = . 7 K Q, Q, ., (K :
Q) = 2, K . . KQ (K: Q) =2, a K, aQ p(x) =IrrQ(a)(x). Q
Q(a) K, Q(a) Q K 10, Q(m) R, m 1. p p (a, c), pb, (a, b, c) = 1.
, , p2 a2 p2 c2, q2b2mc2 Z m , p b, . (a, c)= 1. , 2acZ c= 1 c= 2.
c= 1, a2 b2m Z a, b Z.c =2, a2b2mc2Z a2 b2m 0 mod 4. c =2(a, c) =1.
a . a=2k+ 1. a2 b2 0 mod 4 b2m 1 mod 4. b . b= 2+1, b2m 1 mod 4 m1
mod 4. (*) ___c= 1, R= a+ bm : a, b Zc= 2, m 1 mod 4 R= q+bm2: a, b
. m 2 3 mod 4, R= a+ m: a,Z ( R Z[m]) m 1 mod 4, R= a+m2:a, . , ,R=
a+ q+m2:a,Z m 1 mod 4. R=R. q+m2R, a,Z, a+ 1+m2R. R R. q+m2R, a, .
x, y Z :q+m2=x+ y1+m2. y= x=12 Z. 8k =a+ bm Q(m). a bm k k. k k + k
Tr(k). norm k kk N(k).N(k) R Q(m).R( ) - R, u R u R uu= 1. u u1. R
U(R). 34(i) N(k1k2) = N(k1)N(k2), k1, k2 Q(m).(ii) N(k) = 0 k= 0.56
9. (iii) r R, N(r) Z.(iv) u U(R) N(u) = 1..(i) ki= ai+ bim, 1 i 2
.(ii) k Q(m). k =0, N(k) =0. , N(k)= 0 kk= 0 k= 0k= 0. , k= 0, k=
0. N(k) = 0, k= 0.(iii) r R, IrrQ(r)(x) = x2 Tr(r)(x) + N(r). N(r)
R.(iv) r R N(r) = 1, r r= 1, R R, r r, R. , r U(R). , u U(R), R u,
uu= 1 N(uu) =1 N(u)N(u)= 1. , (iii) N(u), N(u) Z. N(u)N(u)= 1 N(u)=
1 ( Z 1 -1). 15 1. U(Z) = 1, 1.2. R Q(1) =Q(i). 1 3 mod 4, R= a+ i
: a, Z := Z[i]. Z[i] Gauss. Z[i]. a+ i U(Z[i]), N(a+ i)=a2+ 2. a +i
U(Z[i]) a2+2= 1, a2+2= 1, a2, 2> 0. a2+ 2= 1 a2= 1 2= 0 a2=
0
2= 1. U(Z[i]) = 1, 1, i, i.3. Q(5). 5 3 mod 4, Q(5) Z[5] = a +5
: a, Z. a +5, a,Z, a2+52= 1, a2+52= 1 ( a2, 2> 0)., a= 1 = 0. ,,
U(Z[5])= 1, 1=U(Z).4. kQ(m), m , Tr(k) N(k) . IrrQ(k)(x) = x2
Tr(k)x+ N(k).575. k Q(i). N(k) Z, k ;34(iii) k Z[i], N(k) Z.
(iii)34. k= a+ i, N(k)= a2+ 2. , : a2+ 2 Z, a, Z; k=3+4i5 N(k)
=9+165= 5. (iii). 9 1. :k1=1134, k2=10+352, k3=i+552+373.2. a1, a2,
a3 x3+ x2+ x+ 1 ai+2, aiaj, (1+2)ai, 1 i, j 3.3. ;2 + i, 2
+35,q+32,17 +19,3+54,1+232.4. K= Q(m) , m -k1, k2 K. k1, k2 d(k1,
k2) d(k1, k2) =k1k2k1k22 d(1, m) d(1, 1+m2).5. 0a Q(m) , N(a) >
1.6. 1+72a+72Q(7), a, .7. Z u Z(m), u .8. a 0b 0Q(m)a , a N(a)
N().58 9. 35 m R -Q(m). m 1m 3, U(R) = 1, 1. m=1, U(R) = 1, 1, i,
i. m=3, U(R) = 1, 132, 132.. m= 1 2(15). m= 3, 3 1 mod 4. k R,
k=a+32. N(k) = 1 a2+ 32= 4 a2+ 32= 4., U(R) = 1, 1, 1+32, 132,
1+32, 132., , m 1, 3. m 2, 3 mod 4 a+ m R, a,Z, N(a+ m)= 1 a2+ 2= 1
a2+ 2= 1 a= 1 = 0, m > 1., m1, 3 m 1 mod 4. k=a+m2 R, a, , N(k)
= 1 a2+ m2= 4 a2+ m2= 4 a= 1 = 0, m2 22 2. , m 1, . , (. Ribenboim,
"Algebraic Numbers",1972, p.132). 36K =Q(m), m. UK( ) 1, 1 (), 1, 1
, () . , - Q(m). :m2 mod 4m3 mod 4 u =a+ mQ(m) u= 1. u, u1 . u=a+
m, u, u1 = a m, ( ) 1. = 1= a1 + 1m n=n=an+ nm, n+1=a1
n+ an
1., 1 0 221= 2. 1=a1= 1 = 1 +2. m= 3, 1= 1 a1= 2, = 2 +3. m= 6,
= 5 + 26 m= 7, = 8 + 37. m 1 mod 4 = 1=a+m2, a, a, . N(1) = 1 a21
m21= 4 m21= a21 4. 1 > 0 m21= a21 4... m= 5, 521= a21 4, a1= 1
1= 1, 1=1+52. , m= 13, 1=3+132. 9 R . r R , R, r = a, a, R, a U(R)
U(R). r R , p a, a,, p a p . , Z, - . R . 16 1. r Q(m). N(r) = p, p
, r . : r= a, a, Q(m) N(r) =pN(a) =pN(a)N()= p. , 34(iii), N(a),
N() Z. , p , N(a)= 1 N() = 1, a Q(m), r Q(m).2. p Z[i]; p Z p . p
Z[i] N(p)= p2 1. p Z[i]. p= (a+ i)(x+ yi)pZ[i]. p2=N(p) =(a2+
2)(x2+ y2). pZ[i]p= (a+i)(x +yi) , a2+2 1 x2+y2 1.p= a2+2= x2+y2.,
p , p= a2+2.60 9. , p =a2+ 2a,Z. p =(a+i)(ai) N(a+i) = a2+2= p1,
a+i , a i . p . , p Z[i] ... 2= 1 + 1, 5= 1 + 22, 13= 22+ 32 Z[i].
7,11,19 Z[i].3. 5 Z[2]:N(5) =25 1 5 Q(2). 5 =(a+ 2)(x+ y2) N(a+ 2)
1, N(x+ y2) 1,a222 1 x22y2 1. N(5) = N(a+2)N(x +y2) 25 = (a2 22)(x2
2y2). a2 22= 5. :0,1,4,5,6,9. :0,2,8. a222= 5, a222= 5 a2 22= 5. a
5 22 0. , , 5 a5 . 25 (a2 22), , N(x+ y2) = 1.4. Z[6] , 23 =6(6).
2, 3,6, 6 - 23 . Q(6) Z[6], 6 2 mod 4. , - U(Z[6]) = 1, 1. 2 3
=6(6) Z[6], 6 6. 2 =(a+6)(x+y6) 2Z[6], a, , x, y Z. N(2) =(a2+
62)(x2+ 6y2) a2+ 62= 1 x2+ 6y2= 4 a2+ 62= 1 x2+ 6y2= 4 (1) a2+ 62=
4 x2+ 6y2= 1 a2+ 62= 4 x2+ 6y2= 1 (2) a2+62= 2 x2+6y2= 2 a2+62= 2
x2+6y2= 2 (3). (3) , a, , x, y Z, (1) (2) a+ 6 x+ y6 61 Z[6]. 2 .
3,6, 6 -. 2,63,6 -, 2 6 3 6. , 23 =6(6) 6 , Z[6] - . 1801 Gauss :1.
K= Q(m), R .2. m= 1, 2, 3, 7, 11, 19, 43, 67, 163 - R . 2. 1966
Stark Baker, 1. . : 37Q(1),Q(2),Q(3),Q(7),Q(11) , R . 38 Q(m) m=2,
3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 R- . R , (
):m= 38, 41, 43, 46, 47, 53, 57, 59, 61, 62, 67, 69, 71, 73, 77,
83, 86, 89, 93,94, 97. 39 R - . 40 R -K=Q(m): m=5, 6, 10, 13, 14,
15, 17, 21, 22, 23, 26, 29, 30, 10, 15, 26, 30.62 9. [1] . , , ,
1988[2] E. Grosswald, Topics from the Theory of Numbers,
Birkhauser, 1988[3] H. E. Rose, A Course in Number Theory, Oxford
S.P., 1996[4] I. Niven & H. Zuckermann & H. Montgomery, An
Introduction to theTheory of Numbers, J. Wiley, 1991[5] R. Molin,
Fundamental Number Theory withApplications, CRSPress, 1998[6] P.
Ribenboim, The Book of Prime Numbers, Springer, 1988[7] P.
Ribenboim, My Numbers, My Friends, Springer 2000[8] P. Ribenboim,
The Classical Theory of Algebraic Numbers, Springer-Verlag, 2000[9]
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1994[10]
G. Tenenbaum & M. Mendes Frauce, The Prime Numbers and
theirDistribution, AMS, 2000[11] . & . , , , 1991[12] . , - , ,
- ..., 2008[13] A computational Introducion: Number Theory and
Algebra, Cam-bridge, 200563
