Pe^aKTopH: Khib o 2003

Hav^ionajihHa
AnadeMiH
iHcmumym
BUJIMB
nayn
MameMamuKU
HAVKOBOrO
r.
HA
VKpamu
AOPOBKV
B O P O H o r o
CVHA
CHV
HA
VKV
Pe^aKTopH:
F.
Cuma,
A.
tOpauKiecbKuH
Incmumym
Khib
MameuamuKu
o
2003
..
MameMamuKa
FoAoeHUu
A . M .
-ma
CaMOHJieHKO
MameMamuKu
Ha%ioHaji'bHOi
ripaixi
aacmocyeauHn
pedaKmop
iHcmumym
Ku'ie,
vi
AKadeMii
nayn
VKpamu
VKpai'ua
iHCTHTyTy
HauioHajibHoi
MaxeMaTHKH
AKa;],eMii
nayK YKpaYHH
T o m
48
K O H r p y E H u ; ! !
rany
B O P O H o r o
HHCEJI
BEPHyJIJII
jiJi^
IIlTe4)aH nopy6cKi '
y 1890 poi;i r.O.BopoHHH y C B O I H nepmiii, me CTyAenTCbKift, po6oTi
AOBIB, mo AJiH 2m-ro Miicjia BepnyjiJii B2m = P2m/Q2
HO npOCTHMH P2m i Q2m > 0, AOsijIbHOrO MOflyJIH N i KOMCHOrO 6,
BsaeMHO npocToro 3 A'^, Mae Micu,e KOHrpyenmH
»=i
N
(mod N).
U,H KOHrpyenmH i pisHOManiTHi MOAH^JiKauii xa ysarajibHenHJi i i B H HBJiHioTb {JjyHAaMCHTajibHi apH^JMCTHMHi BJiacTHBOCTi HHceji BepHyjiJii. y nponoHOBanift CTaTTi ^ano O F J I H A pisHHx 4)opM, S K H X H E 6yBajio ACBefleHHH xeopeMH Boponoro, i HaMiHeHO AeJiKi nanpjiMH 'ii
sacTocyBaHb.
1. B c T y n
CKpisb HHJKHe nepes Bmix), mm 6 Z*, ^ nosHanaexbCH m-ft MHoroHjieH
BepnyjiJii, ^ mo BHsnanaexbCH posKnaAOM y XBipHHH pap,
exp(z) - 1
'
ml
m=0
' ^ocjii^lHceHHH npoBo;iHJiHCH 3a ni,iiTpHMKH ArcHTCTBa cy6cHjiiH HecbKoi Pecny6jiiKH,
FpaHT # 201/97/0433.
^ N, Z*,Z 03HaMaiOTb BiAnoBJAHO MHO»cHHy AOflaxHHx, HeBifl'eMHHx i Bcix U I J I H X HHceji, a
Q, R, C - Bi/jnoBi/jHO MHO>KHHy paiiioHajibiiHx, A I H C K H X i KOMnneKCHHx M H C B J I .
^ 111 MHOroHJieHH noxoAHTb Bia 5lKo6a Bepnyjuii, H K H H me nepefl 1695 poKOM SHaftmoB
(j^opMyjiy ;uia cyMH (m — l)-x cxeneiuB nepuiHx k HaxypajibHHx HHceji. lien pesyjibTax M O >KHa SHaflTH B Ars conjectandi [4] (c.94-99), ony6jiiKOBaHOMy nocMepxHO. Hacnpaafli 5IKO6
BepnyjiJii poarjiaflaB jiHuie MHoroH.ieHH BepnyjiJii 3 itijiHMH 3HaHeHHHMH apryMCHxy. HaneBHC Ejijiep [19] 6yB HepuiHM, xxo CKopncxaBca H H M H aiin A I H C H H X SHaneHb apryMenxy.
SaysaacHMO, mo nasBy MHOroMJieini BepnyjiJii, 3 BHKOpHCxaHHHM inmnx nosHaneHb, yBiB
[75] flosecj) JliOflBir Paa6e (1801 - 1859) (AHB. xaKO« [78]). Paa6e 3BHHattH0 3apaxoByK)Xb
flo mBeiiuapcbKHx MaxeMaTHKis, 6o ain npo>KHB xaM 6i.nbmy nacxHHy CBOro H<Hxxa. OflHaK
B I H HapoflHBCH B Bpoflax, HKi na MOMenx iioro HapoflaceHHH E X O A H J I H ^ O yropcbKOi H B X I T H H H
ABCxpo-yropcbKoi iMnepil, a H H H I HaJie>Kaxb yKpaini.
48
KoHzpyemijit muny Boponoeo dAsi hucca BepnyAAi
a Bm = BmiO), m 6 Z*, e m-xe M H C J I O BepnyjiJii * y S B H M H H X K H H I napHHX cy4)iKCHHx nosHaMeHHHx ^. Ueft posKjia^ HerattHO npHBOAHTb jxo H B H O I 4>opMyjiH
(2)
= ^ r
/fc=0 ^
x-'^^fc.
^
I3 (1) BHHJIHBae piBHiCTb
(3)
Bm(x + I) - Bmix)
mx"'-'^,
meZ*,xeR,
HKa Aae o^ny 3 HaftcljyHAaMeHTajibHimHx BJiacTHBOCTeli MHoroMJieHiB BepnyjiJii
(4)
y
^m-i ^ BmiN) - Bm^
m,iV = 2 , 3 , . . .
Ufl Ba>KJiHBa TOTO>KHicTb Bi^irpae KJiiOMOBy pojib y noHBi MHoroMJieniB BepHyjiJii B 6ijibmocTi HacxynnHX pesyjibxaxiB. I3 (3) BHnjiHBae, mo Bmi^) —
BmiO) AJiH m > 2. IIosaHK Bq = 1,5] = - 1 / 2 , xo 3 (2) EHnjiHsae peKypenxHa dpopuyna pjia M H C C J I BepnyjiJii, HKy CHMBOJiiMHO MOHcna sanncaxH y
BHrjiH^i
iB + 1)^" = S"^,
m > 2.
SBIACH
BHnjiHBae pauiionajibHicxb
MHCSJI
BepnyjiJii ^. BopoHHfi
TBepAJKeHHH 1. ^Kw,o B2m = P2mlQ2m,
de P^m i Q2m
^
AOBIB
[112] **
> 0 - esacMno
npocmi hucau, mo^
(5)
(62--l)P2„ = 2m6^-ig2™X;«'""'
s=l
^
(mod N)
dAJi doeiAbHozo ModyAH N i b, eaacMHO npocmoeo 3 N.
KoHrpyeHijiH B xaKOMy BHrjiH^i HaBe^eHa, HanpHKJiaA, y KHHrax YcneHcbKoro i Xicjiexa [99] (po3A- 9, c. 261) M H AftpjieH^a i Poysena [33] (po3A- 15).
^ Ilro HasBy aanponoHysaB ae Myanp y 1730 p.
^ 3ayBa>KHM0, mo HanpiiKiHui XIX - nonaxKy XX ex. nm HHCJiaMH BepnyjiJii posyinijiH
HHCJia hm = (-l)"'"'"^B2m, m = 1,2,
IIpH o6roBopeHHi aeaKHx BJKHBanHX noaHaneHb A J I H
HHceji BepnyjiJii ne yHHKnyxH nocHJianb na [2].
€ He oyTKe. spynna 4)opMyjia 3,jin ix o6HHCJieHHa. Eftjiep [19] O 6 H H C J I H B HHCJia BepnyjiJii
AJisi m < 30. Bin riHaiimoB, HanpHKJiaA, mo B30 = 8 615 841 276 005/14 322. V 1842 p.
SHaneHHa HHce.n BepHyjini Sy.nH BiaoMi ^ J I H m < 62 [71]. Haraaaeino, mo B2m+i = 0 ansi
m > 1. BejiHKi xafijiHui iHceJi BepnyjiJii CKJiajiajiHCb 3BaacaiOHH na HHCjieHHi i'x aacxocyBanHa. riocHJiaHHH na inmi panni pe3y.iibxaxH B u,iH o6jiacxi A H B . y [55[, a npo o6MHCJiK)BajibHi
MexoAH i po3uiHpeHHa nonepe/inix Ta6.nHHb - [47|. Po3KJiafl na M H O » C H H K H HHcejibHHKia
B-im AJia m < 30 Mowcna 3HaHXH B (114).
' ^ H B . 3anHCH B ftoro mofleHHHKy [113] noMHHaiOHH 3 4 rpyflna 1888.
" 3ayBaMCHM0, mo opHrinajibHefloaeaeHHaBoponoro CHHpajiocb na npaBHJibHe xBepfl»eHH H P-zmN = Qzm 13^=1
(mod N'^), o6rpyHxoBaHe, oflnaK, X H S H H M H MipKyBaHHHMH. Jlexajii AHB. B [111].
[xj osHaHae Haft6i.ibuie uijie H H C J I O , H K B ne nepeBHmye x.
49
RlmeipaH UopyScbKi
KijibKicTb ny6.iiKauiH.
s'HBjiHiOTbCH HHCjia a6o MHoroHJieHH BepnyjiJii, npoC T O riraHTCbKa. JXna. yxoHHenHH BiACHJiaeMO HHTana
[14]. 3 Hamoi T O H K H
3opy noHBa i'x y U M X ny6jiiKauiMX BUKJUdKaua 'ix rjiH6oKHM 3 B ' H 3 K O M 3 xeopieio KpyroBHx U O J I I B . I3 pisHOMauiTTH sacTocyBaub Muceji BepuyjiJii B ubOMy
uanpHMi, mum B H H H K me no Boponoro, sra^aiiMO Jiuuie Hacxynui, pasoM is
fleHKHMH BH6paHHMH pesyjibxaxaMH uoAajibuioro 'ix posBuxKy.
MyjibTHnjiiKaTHBHa cJjopMyjia PaaSe. IXm, Bi^KpHxy PaaSe [75], Ba>KJiHBy
B.xacxHBicxb MHoroHJieHiB BepHv.ijii MO»<Ha BUBecxu 3 (3):
JJasi doeiAbHozo diucHozo nucna x i VjIaux m > 0, N > 1
N-l
t=o
•
.
^ N
MojKHa noKaaaxH 3a X H X >Ke npHnymeub, mo
(6)
:
Bm{{x})=N^-'J2Brn[\^\
fle {•} - APo6oBa Macxnua nucjia. BuHBjiHexbCH, mo u,e Ay^Ke cJjyHAaMeHxajibHa
BJiacxHBicxb MHoroHJieniB BepuyjiJii.
SnaneHHa ^3exa-4)yHKLi,ii PiiviaHa. Eiijiep [19] suaHuioB, mo
P)
« ^ ™ ) - E ; ^ ^ l i i t " n=l
^ '
BHKopHCxoByiOHH 4)yHKu;ioHajibHe piBUJiHHH pjiH A3exa-(|)yHKiiii'i, MO>KHa
cxH BajKjiHBy 4)opMyjiy
(8)
^ C(-m) = - ^ ^ ,
m = 0,1,2,...
B H B B -
rn + I
OcxauHift pesyjibxax ysarajibuioexbCH uacxyuHHM H H H O M . 5lKmo g{n) - Busnanena na Z KOMnjieKCHOsnaHHa nepioAUHua cJjyHKuiji 3 nepioAOM / , xo BiAuosiAua L-(J)yHKU,iH AaexbCH (4)opMajibHHM) cxeneueBUM P H A O M
a ysarajibiieni
(B
L(,,.,) = f ; ? M ,
n=l
posyMiHui .neoiKx-ibAxa [58]) HHCia BepuyjiJii
g(a.).?exp(az) _
h
~
P H A O M
r>
-»'^-)-'
O A H C 3 HaHBa>KAHBimHx sacxocyBaHb L-cJjyHKuiii L{s,g)
- BHuaAOK, K O JXH 9 = X e xapaKxep. 5lKmo x npHMixMBHuii xapaKxep ffipixjie no MOAyjno
n 3 KOHAyKxopo.M . / ' Y , xo L-c})yHKuiH Jlipixne L{s,x) BHSHanae roAOMopcJJHy
10 SavBajKMMo, mo Pi.MaH, xo'ia 1
HC
4aii i i H B H O , BHOHcaB KijibKa OKpeMHx i'i BMna^KiB.
50
KoHspyeHv/ii m/uny Boponozo Sah hucca BepnyAAi
(jDVHKuiio B I A s 3 K(s) > 1. H K U I O % 7 ^ £ - rojiOBHHH xapaKxep, TO Lis^x)
MO»CHa aHaJliTHHHO npOflOBJKHTH A O rOJIOMOp(J)HO'l 4)yHKUii AJIH Bcix s € C.
L{s,e) = C(s) Mae eAUHHH nojiioc nopHAKy 1 npH s = 1 3 J T H I U K O M 1. Po3KJiaA
Eftjiepa
-1
> 1,
pS
Aep npo6irae Bci npocTi MHCJia, noKasye, mo L{s, x) 7^ 0 pna. ^{s) > 1. Bijibiiie
Toro, [58], [35] (c. 11), npn m < 0
(9)
L{-rn.x)
=
-
Bm+l,x
m+ 1
Ba»cjiHBHMH e SHaHGHHH L ( l , x ) - BHnaAOK rojiOBHoro xapaKxepa e yjKe po3rjiHHyxo. RjiH nerojiOBHOro xapaKxepa x MaeMO Hacxynne 3o6pa>KeHHa B C K I H nennoMy BnrjiHAi, OAep>KaHe F O J I O B H H M H H H O M /Jipixjie [13]:
fx
L{1,X)
=
-^f;x(a)log(l-^7;)
a=l1
h
- ^ Z x { o . ) l o g \ l - m
a=l
•
fx
= ^^¥^i-x.
«Kmo x ( - l ) = - 1 ,
Ae G{x) - raj'ccoBa cyMa, x " KOMnjieKCHO cnpH^ena A O X 4>yHKi;iH i
exp(27ri/m) jxna m S N.
=
HKHIO X " rOJIOBHHH XapaKXep [f^ = 1). xo Bn,x 3B0AHXbCa A O 3BHHaHHHx HHceji Bepnj'jiJii (3a B K H J I X K O M Bi,^ = —B\ 1/2), a flKm,o x ~ neroJ I O B H H H xapaKxep
(mod 4) (y u,bOMy nnnaAKy
= 4), Bn,x S B O A H T B C J I A O
— (n + l)£J„+i/2, Ae Ek e fc-xe M H C X O Enjiepa, BHsnaneHe, nanpHKJiaA, C H I B BiAHOmeHHHM
(£; + l)'= + ( ^ - l ) ^ ' = 0,
npHA;>0,
Eq = 1.
Inmi apHcJiMexHnni B.aacxHBO('xi ysarajibHCHHx MHceji BepnyjiJii MO>Kna snaH X H B [9], [21], ^1 [35] xomo.
PeryjiJipHi npocxi MHCjia. Y sB'HSKy 3i C B O I ' M BiAOMHM BnecKOM y AOBeAenna
BejiHKoi" xeopeMH OepMa (BT<D) KyMMep osnanHB peryjiapni npocxi HHCJia p
HK xaKi, m,o ne Ai-^axb nnaio KJiaciB hir^^^) p-ro KpyroBoro nojia Q(^p). V
cynpoxHBHOMy pasi npocxe H H C J I O nasHBaexbCH Heperj'UHpnHM. KpiM xoro Bin
3ayBa>KHM0, mo CuaByxcbKHii ]87l (c.65) BKaaye na nporajiHHH y (JjopMyjuosaHHi xeopeMH 7 13 [9],flaiTOi'6e3 aoBeaenHsi. lie cxocyexbCH i xBepfljKeHHa 3 3 [21] (c.292), a xaKOJK,
srijiHO 3 [90] (c.64), i XBepji>Keiib 3 i 4 3 [20].
51
lUmeipaH
UopyScbKi
[50], mo npocxe p perj'jiapne xo^i n xijibKH xofli,
HHcejibHHK >KOflHoro 3 nnceji Bepnyjijii
AOBIB
(10)
flijinxb
KOJIH
p ue
= 0
(mod p).
B2.a.i....,Bp_3,
inaKuie Ka»cyHH,
(11)
Pl^Q(^p)
TOfli H xijibKH xofli, KOJIH
KoHrpyeHu;ii IIIxay/i,Ta-KyMMepa.
nacxynny BHsnanny Konrpyenniio:
JJ[
KyMMep [52] (c. 371) xaKO>K
Hexau p - nenapne npocm,e hucjio i m EN mane, ui,o {p—\)
k =
0,l,2,...
(12)
^ '
B g ^ ^ B2^+fc(p-i)
2m
2m + k{p - 1)
\ Todi
AOBIB
Oaji
( ^ ^ ^ ^_
11,51 KonrpyenuiiH bimrpae Ba>KJiHBy pojib y Bcix B I A O M H X AOBeflennHx necKinnennocxi neperyjiapHnx npocxnx HHceji. ITosnanHMO
_ / 5,„/m,
[ {Bm + p~^ — l)/m,
HKiu;o (p — 1) I m
y npoxHBHOMy pasi.
^JKOHCOH [38] (c. 255) AOBiB, mo p2m = hm+p~i (mod p) npn p > 5.
OaKXHHHO KyMMep AOBiB 6ijibme:
Hkui,o P - nenapne npocw.e hucao, a m , n G N maKi, ui,o (p — 1) | 2m,
dAH Xi = P2m+iij,-\)
mo
i 2m > n
(13)
A"x, = 0
(modp"),
;:
de A " noananae n-my pi3Huv,H). a6o, B CHMBOJiinnnx nosnaMennHX i B xpoXH sarajibnimiH 4)opMi. fi'"-{l3^~^ - 1)" = 0 (mod p") sa yMOBH m > n,
(p-l)tm,.
UloAO BnecKy 4)on UIxayAxa, xo Bin y 1845 p. AoniB Konrpyenniio, HKa ne
MicxHXb BKasannx BHme o6Me/KeHb na noKasnnKn cxenenin i B CHMBOJiinnnx
nosnanennHX sanncyexbCH xaK:
^ n + m ( p - l ) ( ^ p - l _ -^y ^
Q
(^^^^J
^^^-1^^
p > i / + 2,1/ > 2, n > i^, m > 0.
KyMMep xaKO>K Z I O B I D , mo njioc LO ' I H C J T O aiJiHXb H H C J I O KJiaciB J I H U I K I B p-ro KpyroBoro
nain xoMHO B nepiuoMy CTeHeiii xo/u ii xijibKH xofli, K O J I H piBHO oflHC 3 H H C C J I BepnyjiJii (10)
^ijHiTbOi nap. y 3B'a3Ky 3 U H M sapTO Bra^axH I I O H S X T H indcKcy HepesyAJipnocmi [65].
^•^ y BCJibMH iH4)opMaxHBHiH po6oTi [93] I.III. CjiaByxcbKHH HaB0;iHXb AeaKi ne flywe
BiziOMi pesyjibxaxH (|)OH UIxay.Tia 3 noro MsAvKe HeflocxynHoi po6oxH [96[, nifl B H J I H B O M
HKoi Bin npononye BJKHBaxn H A S B H meopeMa Ulmaydma-KAayaena npo 3HaMeHHUKU HUc
BepnyAAi. meopeMa Ulmaydma npo huccjiimuku i KompyeHV,ii LUmaydma-KyMMepa.
peKo.MeHflauiil M H HaMaracMOCb jioxpuMyuaxHCb Haaajii.
52
Kompyenv^ii muny Boponozo dAsi hucca BepnyAAi
KyMMep [52] TaKO>K A O B I B cxo>Ky KOHrpyenuiio A J I H H H C C J I Eftjiepa E'^{E^~^ —
1)" = 0 (mod p"), 3a yMOBH, m,o m > n i p > 2. Cnijibne ysarajibHeHHH H H X
pesyjibxaTiB f\jisi H H C B J I BepnyjiJii i EiiJiepa M0>KHa snaHTH B [10].
JXenvl Aajibmi ysarajibHenna KOHrpyenriiH IIlTayATa-KyMMepa 6yAe Aano
HH>KHe.
Is sacTocyBaHb KOHrpyeHuiii IIlTayATa-KyMMepa sraAaflMO, mo IsacaBa [34]
(c. 782) AOBiB, mo HKvuo Kpyzoeuu ineapianm fip dodamnuu,
mo
—— = 0 ( m o d p )
t
—— = ~—-(modp)
2m
^
^'
2m
2m + p - 1 ^
^
dAJi deMKOzo m, muKozo, 'mo 2 < 2m < p - 3 - pesyjibTaT, H K H H 6yB B H K O pHcxaHHii y HHCJiennHx nepcBipKax Horo rinoxesH, nepm hitk i"i A O B C J I H . I l i
Konrpyennii xaKOJK BHAaroxbCH K O P H C H H M H y sb'asKy s inbapianxaMH Ap. A
caMe, KO>KHiH neperyjinpniH napi (p, 2m) siAnoBiAae AOAaxna KOMnonenxa Ap
i UH KOMnonenxa > 1 xoAi n xijibKH xoAi, K O J I H cnpaBA>KyioxbCH Asi BKasani
BHine KOHrpyeHi;ii. Y Bcix BiAOMHX npnKJiaAax n,i KOMnonenxn AopisHioioxb
1 (nanpHKAaA, [62] a6o [63]).
Hacxynni ABi Ba>KjrHBi AOAaxKOBi BJiacxHBOcxi H H C C A a6o MHoroHJieniB BepnyjiJii 6yjio AOBCAeno m,e A O Boponoro. Ilepma s nnx onncye snaMennnKH nnceji
BepnyjiJii ([11]
[95], [96] (naparpacj) 8)). HoBCAenHH u;iei 4)ynAaMeHxajibnoi
B.nacxHB0Cxi nnceji Bepnynjii MO>KHa snanxH B 6araxbox po6oxax
Po6oxa
Boponoro ne B H H H X O K . Bin AaB nesajiOKHC AOBeAennn pesyjibxaxy IIIxayAxaKjiaysena i BHKopncxaB iioro, n;o6 A O B B C X H naneAeny nnacHe rinoxesy AAaMca
npo HHcejibHHKH HHceji BepnyjiJii.
TeopeMa m x a y A x a - K j i a y s e n a npo SHaMennHKH. flKUj,o m > 1, mo
B2m = -
. ^
,
X]
"
(p-l)|2m^
.
de cyua 6epem,t>csi no ecix mamx npocmux. uucAax p, uip (p — 1) | 2m.
HH>i<:neHaBeAena Konrpyennin KapjiiAa [7] niAirpae ponb cnojiynnoi JianKH
3 HacxynHOK) rpynoio BJiacxiiBOCxen nnceji BepnyjiJii:
Hkuj,o p'^ip — l)|27n, mo p " diAumb nuceAwuK uucAa B2m + p^^ — 1Konrpyenuiio Kapjiina neAaBHO noKpamnB CAaByxcbKHH (me ne ony6jiiKOBaHo):
Hexau p nenapne npocm.e hucao i n = k[p — l ) p ' ^ ~ \ k,u e N. Todi
p 5 „ = p - l + A;p'^y
(modp'^+i)
a6o
pBn^p-l
+ kp^ujp
(modp''+^),
^'^ HacnpaBjii KjiayaeH xaK i H P ony6jiiKyBaB aHOncoBaHoro flOBefleHHH.
SraflKH npo panimi po6oTH MO>KHa 3HaHxn B [70] (c. 245).
53
UlmecfkM UopyScbKi
de ujp = {{p — 1)! + \)/p
TeopeMa IIlTayA;Ta
no'd'Hauae euibcoHoey Hacmny.
^ J I H
m/iKe, lufi {p — 1) \ i p'^\2rn.
HHcejibHiiKiB. ([96]): Hkiju,o p - npocme hucjio,
mo
i?2m = 0 (mod
p'').
H K ni-ime CjiaByxcbKHH y [93] (c. 71) "Ue TBepA>KeHHH ne npHBepnyjio ysaTH.
OKpeMi BHnaflKH Li,iei xeopeMi-i A O B O ^ H J ™ JItk. CnjibBecxep, ZL>K. AaaMC,
JlyKac xa in. A.A. MapKOB i H O F O ynenb T.Q>. BopoHHH ne snajiH mei' xeopeMH. SoKpeMa BopoHHii, H K H H OAep>KaB i;eii pesyjibxax H K HacjiiflOK CBoe'i
BiflOMo'i KOHxpyeHmi juin H H C C J I Bepnyjijii, BBa>KaB, mo Bin yaarajibHioe xeope.Niy AflaMca." A caMe, B [1] A^aMC cxBepA;>KyBaB (ne naBOAHHH AOBeAeHHn),
mo BiH AOBiB xaKe: Hkiu,o p > 3 - npocme hucao i (p — 1) j" rn, mo Bm/'iTi
c p " n'ijie HUCAO. y xiii >Ke po6oTi A^aMC B H C J I O B H B SAoraji,, mo K O J I H npocxe
HHCJIO p mJraxb m, ajie ne m-xnxb :3naMenHHK 2m-ro nncjia BepnyjiJii B-im^
TO p flijiHXb HHcejibHHK HHCvia B-^nf BopoHHii o/i;ep>KaB HK nacxiflOK si CBoei
K0Hrp3'ennri nosHXHBny sarajibnimy BiflnoBizib:
HacjiiAOK 1.1. Hkuj,o k = Pi^p^' •••Pr'' dijiumb m, ajie dAH dtcodnoso i —
1..... r
Pi — 1 ne dijiunib 2rn.
m,o HuccAbHUK Hucjia B^m diAumbcn na k.
H K y>Ke sasnananocH. npiopnxex Bi^KpHXXH uboro pesyjibxaxy cjii^, Bi^Aaxn
4)on IIIxayATy, H K H H A O B I B |96] nefi pesyjibxax B CKBiBajienxniii 4)opMi A J I H
r = 1. lien pesyjibxax nisiiiuie ,a,y>Ke nacxo nepeBiflKpHBajiH, nanpHK^ia^,
CnjibBecxep [97], .Jlinmiim |61|. I nabixb (I>po6eniyc nepeAoniB [23] (c. 827)
6es >K0AHHX nocHJiaiib pesy.nb'rdT IIIxayAxa y xaKiii 4)opMi: SnaJueHHUK hucjiu
Bm/m, He Micmunib npocm/ux hucc/i, eidjuinnux eid 3Haju,eHHUKa cua/iozo Bm
16
H K nacxynnnii naciiiAOK cBoei Konrpyennii Boponnn OAepjKan xaKe ysarajibiieniiH KOHrpyeHii,i'i UIxayAxa-KyMMepa ^':
HacTii^OK 1.2.
Hexaii. N = P^^po^ • • -pf''
m ~ m,aKi Hamypujibui Hucjia, uifl
2m HC dijiumb otcodne a hucca p, — \. i — \....
2m — 1. Todi dAH kookhozo n = rn (mod ip{N))
m.Q2mP2n
nQ2nP2m
.r, i m a x { a i , a 2 , . . . ,ar}
<
,
- 2 : 3 — " - 2 J 3ayBa>KHM0, mo B [99] (c. 266) sarajibnimi KonrpyeHn,i'i lUxayATa-KyMMepa
(13) BHBOAHXbCH 3 KonrpycHuiV Boponoro. ZIoxennnM sacxocynannHM noro
KOHrpyennii AJia m = 1 6y.;io I'aKe cnocxepoKennH Boponoro ^*:
Hacjii^noK 1.3. Hucaa
N-l
x =
U'l,
a-12Y^t
1=1
^
N
(mod N)
Iiijuc ysaraJibneHHfl MO>Kna :iBaHTn B [25].
'' He •!rajt\'io'4H B poGoxi IHxayjia ' I H KyMMepa.
.Jlepx |60] (c. 483) 3po6nB xaKnii >Ke B H C H O B O K is HesaJieHcnoro aoBeaeHUH oKpcMoro
BHriaflKV KOHrpyenuiV BopoHoro J I J I H iipocxoi'o M0jj,y,iia (nop. i3 noAajibiuH.M).
KoHzpyeHV,ii muny Boponozo dnji hucca BepnyAAi
CKAadammh unooKuny
{a,N) = l.
poae'mKie
KonzpyeHv,ii ax
=
1 (mod N)
Sah
2. B T O , HHCJia KJiaciB i MHoroHJieHH B e p n y j i J i i
mo6 sposyMiTH rjiH6HHy KOHrpyeHu;ii Boponoro, naBeaeMO KijibKa O C H O B O nojio>KHHx (|3aKTiB, HKi noKa3yK)Tb, m;o sraaani BHin;e pesyjibTaxH ne isojibOBani
OAHH Bifl OAHoro.
5IK y>Ke sra^yBaJiocH, SHanenHH L-4)yHKLiii Biflirparoxb snmny pojib y xeopii'
HHceji. O A H H M is xaKHX npHKJiaAiB e Bi^oMa dpopuyna A J I H HHCJia KjiaciB ( A H B .
Hanp. [69] (c. 372) a6o [30] (c. 7)). Y C B O ' I H aHajiixHHHiii 4>opMi Bona SBynnxb
HaK:
TBepAJKeHHH 2. Hexaii K/Q e cninnenne aScAcee poamupennn Q i nexau
Qi^m) - Kpyzoee noAC, uj,o Micmumt K. Todi Bam HUCAa KAacie hx noAJi K
MaCMO
>^hK=
n
^(1'^')'
x\h=^
dee - zoAoenuH xapanmep, n = 2''^'^'^'^tt^'^Rk/[wk\/\dK\) ~ "cmaAU JJipixAc",
a X npo6izae eci xapaKmepu (mod m ) , mpueiaAtni na nidzpyni H zpynu
FaAya G poamupenuH Q(.^m)/Q, 'UJ,o eidnoeidae K azidno 3 meopieto FaAya, a
x! nosnanae npuMimuenuU xapanmep, indyKoeanuu x-
UloAO apH4)MeTHHHoro ra^rpyHXH cJiopMyjm A J I H HHCJia K^aciB, xo B ; « e KyMMep noMixHB, m,o H H C J I O KJiaciB /iQ(5p) p-ro KpyroBoro H O J I H MOJKna no^axH y
BHXJiHfli Ao6yxKy A B O X ^OAaxHHX ^ J I H X H H C C J I
^i,((j(ep)^2,((}{fp)) flfi ^2,Q[^p) =
^Q(Cp+€"')' ^ sarajibHOMy BiinaAKy nenpaEHJibno ( A H B . [69] (c. 410)), m,o Bi^,HomeHHH /iQ(^„,) AO
^^''^^ '^'^^ AObiAbHoro m G N. Facce [30] AaB
xaKy MOAH^'iKau.iro: Hkuj,o po3mupenHH K/Q a6eAeee, i hkuj^o
no3naHae
MaKcuMUAbHe diUcne nidnoAC 3 K, a h'^ e Hozo hucao KAacie, mo h'^ diAumb
hfc. HacxKy
Tenep H K
Bin nasHBae "BiAHOCHHM H H C J I O M KAaciB posmnpeHKH
xaK i
nijii i
K/K^".
3riAHo s ([30] (c. 13) a6o [69] (c. 410)) / i " ^ ^ ) = "^Kmu)^ ^^^^^o m 7^ p'', i
^Q(im) ~ ^^Mim)
m — p". He AOSBOJine sGeperxn sa H H M H iM'n KyMMepa
i y BHnaAKy p'^-ro KpyroBoro H O J I H .
Facce [30] (c. 12) AOBiB, m,o K O J I H K/Q e X ' H B H C a6ejieBe posmnpennn, xo
/
(14)
h], =
QKwll
\
^
\i
axiia)
/
TaK SBani nepiuHH i flpyrHfi M H O / K H H K H ai^noBi^iHO.
55
UJme(paH FIopydcbKi
Qk nosHanae I H A G K C O A H H H U I posiunpeHHH K/K^, w e H H C J I O KopeniB 3
OAHHHui B
a x i npo6irae Henapiii xapaxTepn K ( T O 6 T O X I ( —1) = — !)•
BHpa3 y Kpyr.iHx Ay>KKax e ne m,o iimie. H K - ^Bi^y.^, xoMy OAep>KaHa 4)opMyj[a
He MiCTMTb TpaHCUeHAeilTHUX B C J I H H H H .
flKiu,o K — Q(\/d) -- yHBHe KBaApaxHHHe posninpenHH Q 3 AHCKpHMinaHXCM
d < 0, XO 3 (14) 0Aep»cyeM0 Ao6pe snany 4)opMyjiy
Ae
0<a<\J\
Ae (d/a) e CHMBax KponeKepa, a 6, a npo6iraioxb xi ejieMenxH SBeAenoi CHcxeMH
HaliMeHiuHX AOAaxHHx JiHiuKis no MOAyJiio \d\, ajih H K H X (d/a) = 1 i (d/b) = —1
BiAHOBiAHO. 3ayBa>KHMO, mo u; = 4 A - J I J I Q(-\/—1), w = 6 A J I H Q(\/—3) i w — 2
AJin Bcix iniuHX yHBnnx KBaApaxninnx posiunpeHb Q.
KoHrpyennii, B H K H X hucjio KJiaciB KBaApaxHHHoro nojin BHpa>KaexbCH B
xepMinax H H C C A BepnyjiJii, Bepxaioxb nac A O Komi, H K H A AOBiB
mo
Rp - Np _ ( W(^jj^]y2^
HKmo p = 3
2
~ \:
HKmop = 7 ( m o d s ) ,
(mods)
Ae Rn e HHCJIO KBaApaxKHnnx AnmKiB, a Nn - T H X neAHmKis sa MOAyJieM n,
HKi > 0 i < n/2. OaKTHHHo Komi AOBiB, mo K O J I H n BiAbne siA KsaApaTiB i
Mae BHrjiHA n = 3 (mod 4), T O
/
f2\
Rn-Nn
= 2n
Ae a, b TaKi nncjia > 0 i < n, mo (a/n) = 1 i [b/n) = —1. Bijibme nocHJianb i
AeiBJieft HHTan snaftAe B [69) (c. 409).
Hexaii p - Henapne npocxe H H C J I O , a Q(^p) - p-Kpyrose nojie. ToAi QQ(fp) =
l,w; = 2p i 3 (14) OAep^yeMO, mo nepmnfi M H O > K H H K
^ KpyroBoro H O J I H
Q(^p) MO»cHa AaTH (|)opM3'jioio
l<s<p--2 ^ a=l
AC X e TBipnHH xapaKTep mod p.
BaHAiBep [102], [31] AOBis. mo
h-^^^^ = {-\)^v~'yH-^^'---'y-'p
JI
S,p.+i
l<,s<p-2
(modp'')
A-iH Bcix 1/ G N (npocTe AOBCAeHHH A H B . y [86]). UK) KonrpyeHuiK) MOSKna
posnHAaxH HK ysarajibnenHH (11).
y BHnaAKy p-ro KpyroBoro ncrin Q(Cp) MaeMO
j = Q(Cp + ip^)- KyMMep [53] (c. 479) AOBiB, mo m'o6xiAHOio i Aocxaxiiboio yMOBOio noAiJiBHOCTi
20 Dicksorrs History HI. c. i02.
56
KompyeHV,ii muny Boponozo Sah hucca BepnyAAi
y nepmoro Mno^nni-ca Mncjia KJiacin KpyroBoro nojiH Q(^p), na p e noAiJitnicTb na p OAHoro 3 nnceji Bepnj'JiJii (10). InaKme Ka>KyMH, Heo6xiAHO i
AOCxaTHbO, mo6 na p AlnnJiocb / I Q ( ^ ^ ) . OanaK A J I H noAiJibHOCTi na p Apyroro
MHO>KHHKa / J Q ^ ^ ^ noAiJibnicTb na p nepmoro MHOJKHHKa e neoSxiAHOio, ajie ne
AocTaxnbOK) yMOBOio [51] (c. 486) (nop. 3 Zahlbericht rijib6epTa, c. 377).
BaHAiBep [102], BHKopncTaBmH uett pesyjibxax i KOHrpyeHuiio lUxayATaKyMMepa (12), nepeAOBiB sraAany BHme KyMMepoBy Heo5xiAHy i AOCxaxnK)
yMOBy noAiJibHOCxi nepmoro MHO>KHHKa na npocxe M H C J I O p.
no.nHHeK [72] AOBiB, mo HKmo npocxe H H C J I O p > 2 AiJiHTb X O H H O i> 3 nnceji
BepnyjiJii (10), xo
^<!J(fpn+l)
—r
= 0
.
, ly^
(mod p ).
• •
Bijibme xoro, neoBxiAnoio i Aoctxaxnboio yMOBOio BHKonaHHH ocxanHbOi K O H rpyenuii e icnyBannH xaKoro rn. mo 2m < p - 1 i
^ 0 (mod p)
i
f'^r''-^
^ ^
(mod / ) .
2rn + p - 1
2m
Tyx p"+^-{i nepBicHHH Kopinb ^^..+1 BH6HpaexbCH xaK, mo6 ajih K O K H O X O n
63'.xo {^pn+iY = ^pn. ysarajiBHenHH MO>KHa snaftxH y [81] i [63].
lUipaxani [82] xaK nomnpnB Konrpyenniio BaHAJBepa na BiAnocni HHCJia
KJiacin:
'^m „+i) ^ ( - i ) ( ' ' - ^ ^ / ' ^ 2 i - ^ " ( ' ' - ^ ) / V ^ ^
n
5^P-"+i,xi ^^od pn
^
l<s<p-2,(s.2) = l,
XI
AJIH KO>Knoro 1/ G N, AG X\: p" xapaKxepin xaK SBanoro Apyroro xnny,
BHsnanenHX no M O A J ' J U O p " + ' . Ilpo Ha6AH>KeHHH L{\,x)
Y xepMinax H H C B A
Bepnj'JiJii A H B . [49].
MipiManoB [66] (c. 54) BWBiB is Kpiixepiio KyMMepa [53] nacxynne: hkuj^o
BT'P
ne cnpaedoicyemhCH dAH eunadKy I , mo
(16)
Bp_3 = Bp_5 = Bp^7 = Bp-g = 0
(mod p).
Ueft pesyjibxax y3arajibmoBa,aH pisni aBxopH. HanpHKJiaA, MopimiMa [68]
posiuHpHB MOKi inABKciB A O Bp-\\ 3a yMOBH 20 579 903.75 571 ^ 0 (modp)
Hasixb AO Bp-13.
JXajihiue npocyBannn B ubOMy nanpHMi MO^na snaftxH
B [40]. IIoxiM BanAinep [101] A O B I B . mo, hk'uj,o BT0 ne cnpaedotcyem-bCH dAH
eunadKy I,
mo
(17)
= 0 (mod p^)
AJI5I
m = {p - 2i)p + 1, Ae i = 2,3,4,5,
mo nicjiH SBCAenHH no MOAyjno p Aae (16).
3aB^aKn BiiiOMHM (HaHpHi<.nai;. ii [77]) o6HHCJiiOBa7ibHHM Me>KaM, o6Me>KeHHH na p y
BHrjiHjii KOHrpyeHuifi xyx MO/Kua BnnycriiTii.
57
nime(paH
UopySchKi
PeayjibTaTH i^boro Tnny nepe^SaHaiOTb KonrpyeHitii ajih HHCeji BepnyjiJii no
MOflyjiK) p^. TaKi Konrpyennii o^ep^ajia JleMep [56]. Bona AOBejia, 111,0 hkiho
( p - l ) t { 2 m - 2 ) , mo
(18)
( 2 2 - _ i ) | ^ ^
Y
{p-2a)2-i
0<o<p/2
(mod/),
0<o<p/3
(22m _ 1) (22'"-! + 1)
(20)
^
=
=
Y
(P-4a)'"^~'
0<a<p/4
(g2m-l ^ 32m-l + 22"^"! _ l )
(21)
(22)
=
,
=
Y
(P-6a)"""'
0<a<p/6
2 2 - | ^ p ^ ^
(modp2),
(modp2),
5 ; (p-2a)20<a<p/2
p > 7,
(mod/).
BaHAiBep [103], [108] (c. 572) AOBin Hacxynne: sikui,o
(1) dpy?.uu MHootcHUK
J HUCAa KAacie noAH Q(^p) e npocme hucao p
i
(2) 'Jtcodne 3 HuceA BepnyAAi Banp '^•^•« " = l5 2 , . . . , (p — 3)/2 ne diAumbca
nap^,
mo BT0 cnpaedotcyemhCH (dAH o6ox eunadme I i II).
MopimiMa B H C J I O B H B Hacxynnnft 3AoraA: hkui,o Bn = 0 (mod p) dAH n <
p — 1, mo HUCAO BepnyAAi B^np ne diAumhCH na p2.
Fyx [29] AOBiB, mo p\h<s^(^^^)
n
XOAI
fl
X I A B K H
TOAI,
K O A H
(S2m£?2m)=0 (mod p).
AnajiorinHHii peayAbxax A J I « nncAa KAacin 6/-ro KpyroBoro H O J I H OAepjKaB K A C 6ox [46]. yaarajibHeHHH pesyAbxaxin uboro xnny na nonni KOMnjieKcni a6ejieBi
posmnpeKHH MOJKHa anaflxH y [87], [88].
JleonoAbAx [59] Aonis: H K H I O nenapne npocxe H H C A O p AJJmTb M H C J I O KAaciB
AiiiCHoro a6eAeBoro H O J I H , ajie ne AiJiHTb cxeninb nOAH, xo AGHKHfi Ao6yTOK
ysarajibneHHx MHceji BepnyjiJii nepexBopioexbCH B nyAb no mod p.
noAJJitnicxb MHceji BepnyjiJii i cnopiAnennx M H C C A saninae TaK0» piani M O AHc^iKami BT<1> ( A H B . [12] i x.n.).
58
Kompyewnii
muny Boponozo
Saji hucca BepnyAAi
MipiiwaHOB [67] sacTOcyeae MHoroHjienH EiiJiepa npn o6HHCJieHHi cyMH
p-2_2P-2^3P-2
±yP~^ y KopoTKOMy AOBeaeHHi /ionoMi>KHOi KOHrpyenuii,
BHKopHCTaHOi Bic{)epixoM AJif! ;iOBeAeHHH ftoro KpHxepiK) 2^'^ = 1 (modp^).
BaHAiBep [106] A O B I B , mo y BTO neMae pimenHH y BHnaAKy I , HKmo M H C J I O
EiiJiepa Eps ne A I J I H T B C H na p.
Fyx [28] A O B I B xBepAJKennn. noAi6ne pesyjibxaxosi MipiManoBa, a caMe:
TBepflHceHHH 3. Hkuj,o p - npocme hucao t zona 6 odne 3 hucca Ep-3, Ep^^,
Ep-7, Ep-(j, Ep-n
diAumhCH na p, mo x^^ + y^P = z"^^ ne Mae po3e'ji3Kie y
v,iAux HUCAUX x,y,z, esacMHo npocm,ux 13 p.
Lti pesyjibxaxn BanAinepa i Fyxa c])aKXHMno AaJin H O U I X O B X Kapjiiuy [8] jiflu
oanaMennfl H O H H X X H peryjinpnocxi B I A H O C H O M H C B A Eftjiepa: Ilpocme hucao p
E-pezyAHpne,
HKWp eono ne dijiumb oicodnozo 3 hucca E2, E/i, • • •, Ep—3, — i
Bin AOBiB, mo e necKinMenno 5araxo ne £^-peryjiHpHHX npocxnx nnceji.
3a AeHKHMH pesyjibxaxaMH C X O C O B H O innapianxiB iBacaBH A, ^ 4p-ro KpyroBoro no.xH i ix : 3 B ' H 3 K V 3 ^E-peryjinpnicxK) niACHAaeMO MHxana, nanpHKJiaA, AO
[17].
EpHBOJi [15] no6yAyBaB saxcuibny xeopiK) neperyjinpnocxi npocxnx MHceji
y xepMinax ysarajibnennx MMc:eji Bepnyji.ii. J J I H npHMixHBHoro xapaKxepa x
Bin osnannn H O H H X X H x~pery„aHpnocxi npocxoro MHCJia p B xepMinax 3'3arajibnenHx M H C C J I BepnyjiJii Biy.B2,^
Bin AOBiB, mo ajih A S H K H X nerojiOBHHX xapaKxepiB (nanpiiKAaA AJia nenapnHx) e HecKinnenno Saraxo ne
X-peryAHpnnx npocxnx Mnceji.
Ocxannboio 3a nacoM, ajie ne Menm Ba>KAHBOio, e AHBOBH^na "napiaana"
4)opMa Kpnxepiio KyMMepa (11), Aana Ep6panoM [32] (oGepnene A O nei xnepA»ceHHH AOBiB Pi6ex [76]). ^oBeAennn xeopeMH Ep6pana sa AonoMororo K O H rpyenu,ii Boponoro A H B . B [33] (rjiaBa 15, §3). EpnBaji [16] AOBiB ysaraAbnennn
xeopeMH Ep6pana, BHKopncxoByiOMM (cepcA inuioro) AoneAeny y [87] Konrpyenniio Boponoro A J I H ysarajibnennx M H C C J I Bepnyjijii i CBoe BJiacne H O H H X X H ne
X^peryjiHpHHX npocxnx M H C C J I .
3. p-'dfl^uHui
L-4)yHKLi,ii
CniBBiAHomennH (12) MO>KHa ysaraJibnnxH ( A H B . nanp. [33]) xaKHM M H H O M :
Hkw,o p - npocme hucao i 'in € N muKC, w,o (p — 1) \ mo dAH m =
n (mod v?(p''))
( l - / " ' - M f ^ ^ ( l - / " - ^ ) f ^
im
in
(modp'').
Ocxanne cniBBiAnomenHH MO>KHa B S H X H sa ocnony p-aAHMnoi' nenepepBHOCxi
snaMenb 4)ynKn,ii (1-p~*)C(.s) npn BiAnoniAnnx BiA'eMnnx mjinx snaMennnx s.
HaflBnicxb eflAepoBoro MnoKHHKa ( l - p " " ) BiAo6pa>Kae 3araAbHHH npHHi^nn: y
p-aAHMHHX anajiorax KOMnjiCKcnnx (JiyKHnin xpe6a no36aBJiHXHCb p-MacxHnn.
KySoxa i .JleonojibAX [49] snaHinjin p-aAHMni anajiorn KJiacHMHHX L-4)yHKmH.
3naMeHHH L(—A;,x) AJIH k = 0 . 1 , 2 , . . . e: uijii ajire6pHMHi nncjia, i xoMy ix
LUme^xm UopyScbKi
MO^Ha posrjiHAaTH H K ejieMeirni anre6pnMHoro saMHKaHHH Qp nojia
pauiOHaJIbHHX p-aAHMHHX MHCeJI. OCHOBHOIO Iipo6jieMOIO, 3 HKOK) BOHH 3yCTpijIHCH,
6yjia Heo6xiflHicTL xpoxii SMiniiTH sitaHeuHH L(k, x)- Ue 6yjio 3po6.JieHO nacxyHHHM HHHOM.
^ j i H AaHoro npocToro HHcna/J HOKjia^eMO
_
_
~ '
HKIUO p > 1
HKmo p = 2.
Hexaii u - xapaKxep TarixMio./i.'iei)a nopHjiKy (p{q} 3 KOHflyKXopoM q. npHHOMy u}{n) = n (mod q) A J I H
G Z i (rt,p) = 1. H K I U O x " xapaKxep ilipix.ne.
xo p-a^HHHa L-cJjyHKuiH Lp(.s, x) osHanaexbCH H K HeiiepepBHa (JiyHKuin Bin
s € Zp, AJIH s ^ I, HKmo X =
HKa 3a.aoBOJibHHe yMOBy
M l - ^'^.X) = - (1 -
Xaj-(p)p-i)
flJiH KOKHoro HaxypajibHoro HHCJia rn.
3ayBa>KHM0, mo
L,{l-m.,x)
HKmo rn = 0 ,
'
=
(modp—1),
,
, „.
(mod 2),
{l-x{p)p'''-')L{\-rn.x).
AJiH p > 2
„
A-ia P = 2.
Bijibine xoro, anajiorinHo KOMixneKcnoMy BHua^KOBi,
He BAaiOHHCb y ;ieTa.'ii, 3ayBa>KiiMO ri.nbKii. mo MynbXHnjiiKaxHBiiy B.JiacTHBic'i'b Paa6e (6) MHoroH.neHiu B(:'piiy.;i.rii .\io>KHa iiepec^JopMyjiioBaxH xaKHM H H H O M : HKmo JXJia X 6 "LINIj HOKJiacxH
i4f'w^iv»-'i/^„(G)).
Ae {x) [losHanae HaHMcriiiie iKMiiji'KMiie ;iiMCHe H H C ; I O
B
:
•
K.'iaci JiHiHKiB(mod Z ) ,
xo ciM^H {Era''} yxi5opioe xaK iBaiii (BcpnynjiieBi) posno/iijiH na {Z/NZ}.
npHBO/mi'b /lo BimiOBiAHoro iiOHHirH .\iipH, sanaHoi cniBBimiomeHHHM
(23)
.ic c.
lie
i?|,f::'(:r) . . / • ! , f ) ( . r ) - r " ' j E | , f ' ( c - ' : ; ; ) ,
1 pauioiiaTbHc
' I I I C ( ; . ! I I > H I I K i siiaMeHHHK H K O C O BsaeMi-io npocxi 3
.viipii npHBe.no }io xeopii inxerpyBannn, y paMKax
HKOi p-aAHMHi X-cjDynKuiY Mo>Kiia inxepnpexyBaxH H K nepexBopennH Mejijiina,
Bi.iiOMe 3 KOMn„'ieKcnoro Buiia.uv>'. 3a AexariHMH Bi;icH.xaeMO nnxana A O [54|.
Mn .JiHuie BHKopHCxae.Mo (l^opM'i'ii'i-M Eiii.,- J..;IH AeMoncxpauii xoro, H K .vio>KHa
liiiBecxn Konrpyenniio Boponoici i ,],eHKi Yi M0,aH45iKani! 3 My./ibTHnAiKaxHBHOi
B.nacTHBOcxi MHoro^uieniB B('))n\'.M.ii.
N. y CBOio nepry ne
HHCJIO.
HOHJIXTH
60
KompyRHV,n rnuny Boponozo dAH hucra BepnyAAi
4. y a a r a j i b i i e H H H KOi-rrpyeHuii B o p o H o r o
' '
Xona KOHrpyenniji Boponoro BiAo6pa>Kae Ba>KJiHBi apucJjMeTHMHi BJiacxnBocxi 4Hce.n BepHy.iiJii, aarajn^ne BH3HaHHH i"i noxpeSyBajio xpriBajioro nacy.
PiBHO Hepe3 15 poKiB, HanpHKJiaA- y po5oxi .Jlepxa [60] 6yjio AOBeAeno OKpeMHii BHnaAOK Konrpyennii Boponoro A J I H naexKH OepMa
q[a) =
.
P1
nenapne npocxe
P
MHCHO.
p
Jlepx cepeA innioro AOBiB )60| (c. 47.3. 4:)opMy.ia (8)), in,o
(24)
q{a)
(mod p).
=
L V
B H K O P H C X O B V I O H H
BHpas
AJIH
nncna KJiacin /iQ(yq5)bu
4%-i)/2^=-E(!;
Aepe npocxe M H C ; I O BHAy p = 1 (mod 4), /;
a (ly/p) osHanae C H M B O J I ,ne>Kai mpa.
.Hepx raicojK
A O B O A H X B
SHanmoB (c. 482). mo
(mod p),
KBaApaTHHHHii He.JiHmoK (mod p),
na c. 483 (He3ajie>KH0 BiA BnmesraAanoro), mo
p-i
(/=!
-
P J u
B I H
-1
126
. p.
(modp),
p\b.
i i5iA3ua4ae iJ>Ke sraAaiie laciocvBaniiH uie'i Konrpyenu,iV A O posB'HsyBannn
jiiniilnnx Konrpyennin. IHo >K A O ysara.'ibHCiHiiM (24) na AOBijibni MOAyJii, xo Bin
AOBOAHXb anaJioriMHHH pe3\-.;ibraT Jiiimc A J I H BijibHux BiA KBaApaxiB MOAyJiiB
m, AJIH HKHX (c. 487):
III-]
ua.
(mod rn).
va L m J
.IlepxoBi AOBCAenHH nboro pe:iy.:ibrary rp>'iixyiorbCH
XOXO>KHOCXi
ua
q{ua) = q{c)
I'OJIOBHHM
H H H O M
na
(mod p).
ua L P
jie 0 < c < p i i/a = a (mod p). i BJiacxiiBocxi qiua.) = q{u) 4- q{a).
BaHAiBep nepeAOBin Konrpyenuiio Boponoro y KijibKOx pisnnx napianxax.
CxaxxH [100] Micxiixb Ana i'i Bapianxn (6e3 H S O A H H X nocnjianb na Boponoro).
DepmHii 3 nnx na crop. 112 rjiacnxb (nisnimi cxaxxi na uio X C M V - [107[ i
[110]):
()l
KompyeHV,ii muny Boponozo dnsi hucca BepnyAAi
H K niAKpecjiHB J ^ J K O H C O H [38] (c. 258), "KOHrpyenuiH siAirpaBajia Ba»:jiHBy
pojib B MHuyjiOMy niBCTOJiiTxi B o6HHCJiK)BajibHiii po6oTi BaufliBepa xa iumux
uaA rinoxeaoK) OepMa [36], [39], [48], [57], [79], [80], [94], [104], [105], [109]. L[i
pisui aBxopH BHKopHCxoByBajiH aauy KourpyeHuiK) jinn CKJiaAauuH xa6jiHUb ueperyjiHpuHX npocxux Huceji. KpiM xoro, B O H H npoAOB^cyBajiu uepeBipKy npaBHJibHOcxi B T O AJIH Bcix uoKasHUKiB, MeuuiHX 3a xi MH inmi Bepxni Me«i, 3a
AOuoMoroK) po3po6jieHoro Bau^iBepoM nsBHoro KpHxepiro." Baruixac[) [114]
caM i cnijibHO 3 TaunepoM [115] upocyHyB ui oSMHCJieHHH suaMHO Aajii, B H K O pHCX0ByK)MH KOHrpyeHi^ii, noAi5Hi A O (25). HeAaBHi o6MHCJieHHH, cxocoBui U H X
UHxaub, 6yjiH BHKOHaui ByxjiepoM, KpaHAOJioM, EpuBOAOM i MexcauKijia [5],
OAHaK BOHH CHHpajIHCb Ha iHIUHii MexoAFprau [27] nepeBiAKpuB KonrpyeuuiK) Boponoro y BurjiHAi
TBepA,>KeHH5i 7. Hexaii, p > 2 - npocme hucao, a - mane u,iAe hucao, w,o
p\a,
i m > l -
napne hucao. Todi
p-i
ak
(26)
(mod p).
LP J
BHKopHcxoByroMH KOHrpyenniio (26), FpioH uepexBopHB Ao6pe 3HaHHft Kpnxepift KyMMepa '^^ pna BunaAKy I : [Hkui,o (pi{t) = E n = i ^ • ^ • ^ * ~
l , . . . , p — 1 , i t = —y/x, mo (/Pj(i)5p_i_i = 0 (modp)] - A O eKBiBajienxHOro BHrAHAy
p-i
fc = l , . . . , p - l .
E
n
niACxaBHBmn m = p — 1, Fpiou OAepmaB BHAOSMiueHy cJjopMy xoxo>KHOcxi
Jlepxa (24), 6e3 > K O A H H X uocHJianb ua Jlepxa.
Ty caMy K0HrpyeHu;iio, mo B Fpwna, Jinme sa AOAaxKOBoro npHnymeuHH
(p - 1) I m, AOBiB xaKOJK 4 > K O H C O H [38] (xeopeMa 5) p-aAHMUHMH MexoAaMH.
^ J K O H C O H SBepxae yBary, mo (17) i (26) nacupaBAi piBHOCHJibHi.
fljKOHCOH [38] (c. 257) 3ayBa>KHB, mo hkui,o p = 3 (mod 4), mo (26) dae
f
2-
(2\
\P)
(p-l)/2 .
B
(modp),
(P + l ) / 2
^
\PJ
Po6oTa [38] BHHuiJia B 1975 poui.
3ayBaH<HM0, mo pe4)epeHT FpioHa B Zentralblatt cxBepA»<yBaB, mo npHnymeHHH npo
napHicxb m saftBe. IIpocTHH npHKJiaji p = 3,m = 3,o = 2 a6o nopiBHHHHH 3 HaBe/ienHM
HHJKHe peayjibxaxoM noKaayioxb, mo ne ne xaK. noAi6HHM H H H O M , xona y cxaxxi nePt B K H E W O K
He BHKJiKDHeHO, apryMenxauifl FpiOHa xH6Ha A J I H p = 2, H K 3ayBa>KeHo B ]64] a6o H K u,e huvkhb
BHOjiHBEie 3flOBeflSHoipaHiuie ^opuynu (27).
2"* C T O C O B H O aHajiorinHoro KpHxepiio, oxonjiK)K)Horo HHCJia Eftjiepa, A H B . [28].
63
Ulnu.'cfxm nopy6("bKt
de (ajp)
odHUHae cuMCioJi. •Jleotciindpa. 3 B I A C H BHruiHBae, HanpHKJiafl,
mo
B(p+\.)i2 ^ 0 (modp) aoo mo
•
% v ^ ^ i = ~25(p+i,/2
(modp),
- TBepA>i'ceHHji, Bi,aoMe me Komi
i nepeaoBeaeHe Opi^ManoM i TaMapKiHofl [22] (c. 154).
CjiaByxcbKHfl [85] ,aoniB
ysaranbHennH KOHrpyeHuii' Boponoro y xaKifl
4)opMi: 2'''
TBepflJKeHHH 8. Hkwp N > I i a
esucMHO npocme 3 N,
HamypaAbni uucjia, a u^'uie hucao a
mo
„m _ 1
(27)
ak
2-
B„,=2T{aky"-'
^
in
iV
+ {1 -a)Bm-iN
(modiV).
Hacxynui naauAKH BHn.;iiu5aioxb is nonepennboro TBepA>KeHHH:
HacjiiAOK 8.1. flKvip X > O.m > 2. N > I i dAH v,iAO?.o nucna a {a,N)
TOO
mp
(x\ p
^o_Bm[J__Um
m
A'-l
^ 2j2i^,k
+
xr-'
ak + X
N
+ il-a)Bm-iN
= 1,
(modiV).
BH6iipaiOMH a =
Kon/ivKiop Hero.xoBHoro xapaKxepa X- CjiaByxcbKHPi
BHBOAHXb is nonepcAiiboro nac.;n;u\
HacniAOK 8.2. flnwp KondyK'nwp
cmuu 3 N > I i ni > 2, mo
h
nczoAocmozo xapaKViepa x 03aeMH0 npo-
.v-i
(mod N).
2 ^ - f E x ( - ) E ( . / x ^ ^
N
,r=l
k^O
3 o< rannboV KonrpyenuiV ;vva uixxroro MOAynH N — p BHnjiHBae, mo K O J I H
( p . / ) = 1 i / > 1, XO Biii,y p ui.:ic. Bliibine xoro, sa xnx caMHX yMOB MaeMO,
mo KOJiH p''||7n., xo B„i,^/p'' raKO/K p ui.ne (5es o6Me>KeHHh na rn y Buna^Ky
SBHHafiHHX MHceji BepHyj[.;ii).
3 ocxannboV Konrpyenuiy imo/K HuirnnBaK ]85] (c. 124), mo koau Q(\/d) c
y,Hemi.M KaadparnuHHUM n.oACM 3 ducKpuMmaHmoM d, < —4, a N > 1 mane,
Z[nB. <;. 50 uie'i CTarxi.
2'' Ha >Ka.7ib, y cxaxri e fiaraxo Kajx I . K H X I I O M H J I O K .
5IK CKa^aHO B Kinui cxaxri, Kompycuuii uiei crarxi MicxHXb y:5ara;ihHeHH5i KoiirpyeHui'i
Boponoro, DoneptviHbo AOHe;i,eni B [loooiax |41] [44] i [84]. Y [45[ Konrpyeimi>i Aaexbcji 6(!3
Aone.ac'UHH, Jiiimo 3 nocM.)iaHiiJiM. repcji iiiiiioio. na [83j. OanaK KOHrpyeHuiio (:fl)Oi)My.ribOBano B |45] (.neMa 2) y uHr.!inji.
o< raiiui.M iiiipa3o.M iipaBoi nacxHHM e (1 — a"')B,„Ar saMicxb
(1 -a"')B,„_,iV. •
2** riepiiiHH 3 UHX xaKOyK 3ra;i>'<"ibcsi n ]45], a ao xoro B [44] JiJia Aoaijibnoro x. B o6ox
BHna,-y<ax 6e3 AOBeAenna i iiisi .\io,iy.;isi A' =
p - Henapne npocxe H H C J I O .
KompyenVjii muny Boponozo djiH hucca BepnyAAi
mo cuMBOA Kponenepa x{p) = (d/p) = — 1 dAJi Kootcnozo npocmozo diAbnuna
p HUCAa N, mo
HapeinTi 3 ocTanHbOi Koi-irpyeimii 6yjio oriep>KaHO pnA BHpasiB A J I H Bm,x/''^
npH
m = ^-^p'"'^
+ I. a6o (p -
a6o [p - I'jp"-^ + 1
i xapaKxepa xi^) = ( x ( — i i ; o B nosAHaHHi 3 xBepaxeHHHMH, p.OBejxeHHMH B [43] i [45], MO>KHa BHKOpMCXaXIl JXJIil BHpaxeHHH aeHKHX BJiaCXHBOCXeii
noAinbHOCxi HHCJia KJiaciB KBaApaxnHHHX no.niB y xepMinax nijioi" nacxKHH [-J.
IHoAO po3UiHpeHb KOiirpyenuin IlIxayAxa-KyMMepa, CjiaByxcbKHii y [85]
AOBiB
HacjiiA;oK 8.3. Hexau N > I - namypaAbne hucao, N = p^'p"^ . . . p"*^ iiozo podKAud y do6ymoK pisnux npocmux hucca, a m i n ~ mam napni na-
mypaAbni hucau, uj,o rn = n (mod (p{N)) i m a x { a i , a 2 , . . . Or} < m i n { m , n } .
Todi dAH KooiCHOZo esacMHO npocmozo 3 N nenyAbeozo u,iAozo hucau a MacMO
ajn _ 1
2
Hkui,o do m,ozo ok: N
p\N,
(jn _ 1
B,„ = 2
Bn (mod N).
rn
n
nenapne i (p — 1) f m dAH Kootcnozo npocmozo diAbnuna
mo
J
,
m
n
,•
-•
HacxynHHfl nacjiiAOK 6yjio AOBeAeno B [6] 3 BHKopHCxaHHHM inmnx 3aco6iB:
HacAiA,OK 8.4. Hexaii, p" e cmeriiub nenapnozo npocmozo nucAa, a m,n maKi dea namypuAbni HucAa, wp rn = n (mod <p(p'')) i v < m i n { m , n } . Todi
dAH KOOICHOZO p-VflAOZO HUCAU b
=
rn
(modp).
n
HacaiAOK 8.5. Hexau N > 1 ~ namypaAbne hucao, N = p^'p"^ • • -Pr"^ - uozo
po3KAad y doSymoK pianux npocmux, a m i n - mam dea napni namypaAbni
HUCAa, uj,o rn = n (mod ^[N]) i m a x j a i , a 2 , . . . Or} < m i n { m , n } . Todi dAH
KO'jtcHOZO eaacMHO npocmozo 3 N nezoAoenozo xapaKmcpa x 3 KondyKmopoM
f^>0
MUCMO
EllhK = Elhx
m
n
(modiV).
CAaByXCbKHH [87] AOBiB
TBepA:»ceHHH 9. Hexaii x - xapaKmep is KondyKmopoM.
^, P ^ npocme
HUCAO i a - muKC u;iAe hucao, ujp [a, }\p) = 1. Hexaii daAi s = ordp{f^) i v neeid'euni 'niAi hucmm, Bah hkux max{;y, . S } > 1. Todi dAH Kooicnozo m G N
65
LLImecpuH TIopydchKi
ak
= 2 E
(mod p''+').
X{ak){akr-'
k=l
Sonpeua, jikiho p > 2 i x("~l) — ( - I ) ' " , "lo
ak
( X ( a ) a - - 1) ^
^
(mod p^^^).
E
III
Hapemmi, siKXii,o p \ fy^ > l,p > 2 i u > 1, mo
, /xP"
Br
(mod p").
m
Hkw,o p ~2,
IP']
k=l
mo ocmauHH KompyeHii,iji cnpaedotcyemtcji no ModyAW 2" ^.
y [74] KOHrpyeHi],iio Boponoro 6yjio BHBeAeHO 3 MyjibTHnjiiKaTHBHOi BJiacxHBOCTi Paa6e (6). ^OBefleKHH cnnpaeTbCH na anajior Ao6pe BiAOMOi 3 anajiisy
4)opMyjiH AH4)epeHii;iK)BaHHH x"'- :
(28)
mD{m)E^J^^Hx) = mD{m)x'''-^E[^P{x)
(mod M ) ,
a6o
1
c, c
M
mD{m)
Ae c 7^ 1 - B3aeMHo npocxe 3
panionajibne M H C J I O i D{m) - naftMenme
cnijibne Kpaxne snaMennnKiB Koe4>ii;ienxiB MnoroMJiena BepnyjiJii Bm{x). Topi
4)opMyjia Paa6e (6) Aae
Nd-l
(29)
Ae
N-l
E < ' ^ = E--"''<^(--)
x=0
.x=0
= npiivP"'''''''"'^^"'''- OcKijIbKH
c
+
c- 1
2
'
xo npaBa MacxHna (29) naGyaae BiuvmAy
N-l
, iV-1
2 „tt
OAnaK AiBy Macxnny M0>KHa nepexBopnxH A O BHrASAY
. , :
(l_c'"):?i!i
m
29 OKpeMHii BHnaAOK uieT KOHrpyenuii G V A O AOBeAeno y
66
[18]
(xBepAiKeHHa 3).
KompyeHUflt muny Boponozo dAsi huccm. BepnyAAi
jinme
TOAI.
KOJIH
pauionajibHe
HHCJIO
Mae
C
BHFJIHA
c = 1/b, m b ^ I ~ nijie
Lie BHiipaBJiHe noMHJiKy y TBepA^KeHHi O C H O B H O I xeopeMH 3 [74], mo
KOHrpyeHuiio Boponoro MO>KHa ysarajibHHXH xaKHM H H H O M , mo 6 e AOBijibHHM
pauioHajibHHM H H C J I O M ^ 1, B3aeMH0 npocxHM 3 A''. H a HenpaBHJibHicxb xaKoro
ysarajibHeHHH BKasajiH MexcanKijia [64] i CjiaByxcbKHM [90], [91], [92].
HaBeAeni BHme cniBBiAHomeHHH pasoM is
HHCJIO.
7V-1
_
Ya'--'=NBm-i
a=l
-
+ ^^N^Bm-2
^
-
(modN)
AJIH m > liA'^ > 1 HepeAOBOAHXb BHmesraAanHH peayjibxax CjianyxcbKoro,
HKHii y [74] HaBCAeno B "napnoMy" Bapianxi (BHnpaBAeHOMy B A,yci nonepeAHboro 3ayBa>KeHHH)
iV-l
ab
(mod A^),
N
a=l
i B "HenapHOMy"
6-1
B2kN =
E
2k
ab
N
a=l
( m o d N)
AJIH BsaeMHO npocxoro s A^ nncjia 6 i A; G N . IJe npHBOAnxb is BHKopncxanHHM iABH /^JKoncona [38] ao "napnoro" i "nenapnoro" BapianxiB M0AH4)iKau;ii
BanAisepa Konrpyennii Boponoro pjia. N > I i 6 > 1, BsaeMHO npocxoro s
N:
6-1 [vN/b\
(1_62'=)^^262'^-^E E
(mod AT);
^'
v=l a=l
HKmo KO>KeH npocxHH AiJu^nHK p HHC-xa A'' saAOBOJibHHe yMOBy (p — 1) | 2A;, xo
6-1 [vN/b\
E
v=l
E
a=l
= °
H K 6yA0 BKasano, "nanKpamiiM napmiM" ysarajibnennHM Konrpyenuii Boponoro e
Bo,
( 6 - - l ) ^ +
N-l
62'= - 62'=-! 2k - 1
a=l
0.6
N
(mod
N).
HncejibHHK 62'= — 62'="^ MO>KHa saMinnxn na 6 — 1, BHKopHCxaBmn B AOBeAenni Konrpyenmio Yla=i
= E ^ / ( ^ ' ^ ) ^ ' ^ ~ ^ ( m o d A'') (n;eH cnpomennn
Bapianx AOBeAeno B [90], A H B . HH>KHe).
y [89] CxaByxcbKHH sanncan ne y cj^opMi
67
LUme(pa.H UopydchKi
N
ab
rn-l
l-b
+•
N
m
m — 1
NBm-2
N
(mod N),
fle m,b,N G N saAOBOJibamoTb V M O B H m > 1,6 > 1,A'' > 1 i {b,N) — 1.
H K sacTOcyBaHHH Bin OAepjKaB nocHJieHHH xeopeMH JTinuiHUH-CHJibBecxepaHijibcena ( A H B . HH>KHe) i BHnpaBJieHHH Bapianx AOBeAenoro YexapoK) XBepA>KeHHH [98].
CjiaByxcbKHH [90] xaK0>K AOBiB HacxynHHii pesyjibxax:
TBepA>KeHHSi 10. HKwp m, b,N - uamypaAhHi hucaq. b i N - esaeMHO
npocmi i ^ - nepeicHuu Kopiwu b-ao cmencHH 3 oduHuv,i, mo
(30)
m
c=l a=l ^
OKpeMi BHnaAKH uie'i KonrpyeHtiii MO>KHa 3HaHXH B>Ke y [23], a jx^a N = p^,
Ae p
npocxe nncjio, i i A O B I B Yexapa [98]
CnaByxcbKHH BHBiB is nonepeAHboro pesyjibxaxy nocnjiennji '^raAanoi BHiii,e Konrpyennii Boponoro y c|)opMi BaiiAiBepa. II,eii pe3y./ibxax AoniB xaKO>K i Vexapa. OAnaK is nporajinnaMn,
CnpHHHHeHHMH sra/jaHHMH Binne noMHJiKaMn:
TBepA,JKeHHH 1 1 . Hexau rn > I, b > 1, N > I - v,iAi uucAa, {b,N) = 1 i
u,iAe HUCAO g{a) ausHanacm/bCH yMoeauu a = g{a)N (mod b) i I < g{a) < b.
Todi
Bm ^
(31)
rn
= - E Oi^^h""' +
a=l
^-^N
^
(Bm-i
V
+ '^^NBm-2)
^
J
(mod N).
'
SoKpcMa, HKUi^o HC 2\\N i odnoHacrio 6 = 1 (mod 4 ) , mo Oah
A2/C
(32)
_
A''-l
1
—B,,
2k
6 N
= - E
(mod N).
Bin SBepnvB yBary. mo U H K()Hr))yeHniH c{)aKXHHHO e H O B O K ) cJ^opMOio (ysarajibnennHM) K.xacHHHoi KOHrpyenuii Boponoro. Bin noKasaB, mo sepxaiOHHCb
AO ui,Jioi HacTHHH y cniBBiAiiomoiiiii (31), A J I H rn > 1 OAep>KyeMo: ^
-JjrB'n
=
iV-l
1 ab
+
= (-i)'"6'"-i E « ' " " ^
a=l
N
2
Bm-l
V
30 Moxviia 'iBipHTH :s BHiipaB.ieiiiiii.Mii 11 [911.
68
+ '^^^Bm-2N]
(mod A^).
KoH2pyeHV,ii muny Boponozo dAR hucca BepnyAAi
JXjir m xaKoro, mo 2!m i m > 2. ue ^ae
iV-l
b"" - 1
Bm =
m
E
a=l
m-l
ab
+
N
^-^-'^Bm-2N'
(modiV).
JXjih m TaKoro, m,o 2 f m i m > 1, is (31) OAepjKveMO
6-1
N-l
BmN
= 6'"-^
a6
(mod AT),
E
m
mo BunpaBjiHe uacniAOK
1 is [74]. Hapemxi, HKmo ue 2||A'' i OAHonacHO 6 =
— 1 (mod 4), xo 3 (31)mcxaeM0 jxjih 2\in
N-l
b"' - 1
Bm = b""-'
m
ab
E
(mod
N
0-1
N).
Hijibcen [70] (c. 250) A O B O A H X B , ysarajiburoroHH pesyjibxaxu CujibBecxepa
i .JIinmHu;a, mo b''~^^{b'^^ - l)B2k/2k
- nine H H C J I O npn 6 G Z i A; G Z . CjiaByxcbKHH [90] BHBOAHXb is KOHrpyeHuii Boponoro uaiiKpame M O X J I H B C ysarajibneuHH pesyjibxaxy Hi^ibceua, a caMc: Oasi 6 G Z i m G N nauMcnmuM uamypaAhHUM hucaom h. Bar hkozo hucao b^{b"^ — l)Bm/'m - u,iAe, e
h = [log2 m] -I- 1.
A r o [3] noMixHB, mo, d^iKcyiOHH B KOurpyeHmnx Boponoro m i SNiinioionH
n , 0Aep>KHM0 nacxynne uiACHjiennH (Q2m ^ snaMenuHK B2m)'
TBep/i;>KeHHfl 12. Hexau m,n,w
- doeiAbui namypaAbni HUCAa maKi, uj,o
Iw.
p\n
Hkw,o a - uarnypaji'bHe hucao i {a.w) = 1, mo
ka
l<k<w-l
{k,n) = l
p\n
w
(mod
n).
Is imoro pesyjibxaxy Bin B J I B I B nacxynne ysarajibneuHH Konrpyenniii IIIxayAxa-KvMMepa, AOBCAene Opo5eHiycoM [23] (c. 842) A J I H n , HKe e cxeneneM neiiapHoro npocxoro HHCJia:
H a c j i i ^ O K 1 2 . 1 . Hexau n > 3, u. 'tn, a - namypaAbni nucAa i (a, w)
—
I . Hkuj,o
<p{n) - cfjyHK'niM EuAcpa, mo
y ( _ l ) ' c f ^ ^ TTn
k—l
.
2m-l+/c^(n)w^2m+fc^(n)_^) ^ 2 m + M n )
p\n
SonpcMa, HKiu,o p — l \ Sah ecix p\n, mo
k=l
^ ' Pin
69
^ Q
^^^^
UlmeifiaH
UopyScbKi
5IK y>Ke 3iaAyBa„iocb, ,ueHKi peayjibxaxH cnHpaioxbCJi na icHyBanHH KonrpyeHLi;iH no MOAyjiio cxenena npocxoro nHc:.Jia nn, aarajibnime, no MOAyjno cxenenH
naxypcuibnoro nncxa. H a nepmnn norviHA s^aexbCH, mo xana cnpo6a noKpnBaexbCJi XHM. mo KOinpyennia Boponoro sanncyexbca no AOsijibHOMy MOAyjno
A'', H a >Kajib, npaea nacxHna Konrpyenmi' ne Mae Ha.xe»cnoi cxpyKxypn.
y 1974 p. /^jKoncon [37] BnBiB Konrpyennii (18) - (22) is nacxynnoro ysara.ibnennn Konrpyenuii Boponoro:
T s e p f l ^ e H H H 13. Hkiho ( p - 1 ) \m — 2) i b - i^iyie hucao maKc, uj,o
2 < b < p - l , rno
(33)
{b""-l)B2ra
ba
2m- 1
=
p-i
p 6 2 — 2 ^ a ^ 2m-2
LP J
ba
(mod p^)
LP J
a=l
FaHAi, cni.ibHO s Kecy6e i CypanapHna [24] Aonejin nacxynne niACHJiennH
KOHrpyenuii Boponoro.
TBep/^>KeHHH 14. Hexau p
nenapne npocm,e hucao, rn > 2 - u^Iac hucao
mane, uj,o (p — 1) | (2m - 2), a 6 - eaaeuno npocme 3 p u^iac hucao. Todi
(34)
{b^- - 1) Urn
+
-(^^^)p62—2|^a2'"-2
''-^^^^^B2m-2P')
6o
—
B O H H
p-l
2 /2m\
.p .
BiATaK
^ 2m62-i £
ba
^2m-3
V 3 ;
-
(mod p^).
Ip}
BHBejiH s HCi KOHrpj'eHmK) (xeopeMa 1.1):
Hacjii/i;OK 1 4 . 1 . Hexaii p - nenapne npocme hucao, a m > 2 - maKC v,iAe
HUCAO, uj,o (p — 1) I (2771 - 2). Todi
2
^52.mPH
=
E
±!2m-2P j =
(P - 2a)^'-" + m(2'm - l)p2
0<a<p/2
HK
^
{p - ^a)^""'^
(mod p^).
0<a<p/2
spasoK M0H<jiHB0CxeH posmnpenb KonrpyenuiH .JleMep
B O H H
AOBCJIH
Hacjii;];OK 1 4 . 2 . Hexaii, p
nenapne npocme hucao, a m > 2 - mane u,iAc
":urAo, 111,0 [p — I) \ 2). Toch
i:i^'''~^){B,m-^'^^^^^
E
(p-2a)--
0<rt<p/2
E
V ^ /
0<a<p/2
( P - 2 « ) - - + M /
^ '
70
E
0<a<p/2
(p-2a)^'"-^
(mod/).
KoHzpyeHv^ii muny Boponozo Sah hucca BepnyAAi
y [73] iaeio posno^iJiy BepnyjiJii Syjio TaKO>K EHKopHCxaHO ajir noniHpeHHH
/l^HCOHCOHOBoro pesyjibTaTy na BHnaAOK AOBijibHoro TV saMicTb npocToro p .
TaM AOBeAeno, mo "KBaApaTHHHHM" anajioroM cniBBiAHomeHHH (28) e
D{m)mEZH^)
= D{rn)m
Urn - l)x"'-'^E^^'^Hx)
=
- (m - l)!""'^e[^\x)\d
ToAi npaBa nacTHHa (29) Ha6yBae BHrjiHAy
M^).
.
B2{l-c^)-2Bic'z-Boc^z'
E
+
x=0
N-l
+ Y x ^ - ' [ - c z
+
Bi{l-c)],
x-O
flfi z = —[c~^x/N\.
Mo>KHa 0Aep>KaTH poamnpeHHa KOHrpyenmi BopoHoro,
BHKOpHCTOByroHH U,IO TOTOXCHiCTb 3a yMOBH, mO SHOBJ' C =
,
t £ 7j,b ^
1 i 6 BsaeMHO npocTC 3 N. TaKHM H H H O M , H K nacjiiAOK, MO>KHa OAep^Kara
(cepeA inmoro) 6e3nocepeAHe ysarajibHeHHH peayjibTaTy /^jKOHCona (floro 6yjio
AOBeAeno sa AOAaTKOBoro npHnymeHHH k > 4, HKe, H K BiA3HaHeH0 B [64], e
SaHBHM).
TBepA»ceHH5i 15. JJkuj,o (p — 1) { (2m — 2) dAH Kooicnozo
p HUCAa N > 1 i u,iAe hucao b esaeuno
npocme 3 N,
mo
N-l
2m - 1
ba
N-l
Nb
:2m-
N
2m-2
npocmozo
ba
diAbuuna
(mod
N^)
N
a-1
H O T I M , is BnKopncTaHHHM iAeii /Imoncona ysarajibneHHH Konrpyennift JleMep (18) - (22), Moacna AOBecra, nanpHKJiaA,
H E C A I A O K
KOJKnozo
15.1. Hkui,o N - m,aKe u,iAe hucao, iu,o (p — 1) f (2m — 2) dAH
npocmozo
(2 2m
diAbnuKa
• B2m
11
2m
p
=
hucau N
, mo
{N-2a) 2m-1
E
(mod
N').
0<a<N/2
y [64] MexcanKijia npononye npocTimnn i ejieraHTnimHH, onepraH na (4),
MexoA snaxoA^KennH ysarajibnenb Konrpyennii Boponoro no MOAyjno AonijibHoro CTencHH A^". Bin nonnnae s
ba
ba =
N + ra,
0 < r a < N - l ,
N
I, niAHiMaiOHHCb noTiM AO m,-ro cxenenn, 31 OAepjKye (socepeA»yK)HHCb
rojiOBHHM HHHOM Ha BHHaAKy napHoro m)
'^^ Ila i f l e a
n p n c y x H a x a K O K y n e p a i c H O M y AoaefleHHi
BopoHoro
(nop. AOBefleHna A B M H
i B AOBefleHHi O K p e M o r o B H n a A K y x s e p A i K e H H a 9, n e p e A O B e A e n o r o B [16]
71
(AeMa
1).
X)
UlmecfkiH
TBepflJKeHHH 16.
mo
Hkiu,o
N.b,k
nopy6cbKi
- Hamypajibm
•! ,
Hucna,
(N,h) = 1 i m G Z * ,
V
(35)
rn
c^e T i ( i V ) = E a = Y ( ^ " ) • ' " " H V / ^ ' J ' •
5IK HacniAKH CBoro nLuxoAV. MercaHKijia ^aB HacrynHi KOHrpyeHuii no M O Ayjno N i N'^ . riepnia 3 nnx e Menin BiAOMHM BapianxoM Konrpyennii' Boponoro:
HacjiiflOK
1 6 . 1 . J],jiH
[h.N] — l i r n > l
(pi'n _
MaoMO
= 2ri(iV)
( m o d N).
771
3HizicH i\n>K i n m H M BanjinBae:, mo n p n (b.N)
A' nijiHM.
H a c j i i A O K 1 6 . 2 . Hmup
-
-
e nucmdny
(6-^'" - 1)B2,„
16.1
m >2.
+ (b"" - 1)"'^"^
= 2wT\ - rn{2rn
= 1
- l)T2iN)N
HHCJIO
(ft^'" — l)B2m/'rn.
e
mo
~ '^B2m-2N'
^
( m o d A^'^).
I l p o inmHH 3B'>t30K Mi>K .MVJibXHiiJiiKaxHBnoio xeopeMOK) Paa6e i Konrpyen-
uifexo Boponoro A H B . B [26].
no;jaHK AGHKi 3 u,H'roBauiix po6i'r 5y.xH neAOCHxni pun aBxopa, A^siKi pe3y.;n>Taxn, MO>KjrHBO, npouyini'ni. M n BH6aHaeM0Cb nepcA anxopaMH. 3 inmoro
6oKy. HeMO>KJiHBO BK.?noHHXH Bci 3acxocyBannH Konrpyenm'i Boponoro nasixb
x i , npo HKi HuiJiocb y BcxynnoMy po3Ai./n uie'i cxaxxi. 3a AOAaxKOBUMu AexajiHM H BiACH.iaeMO Huxana A O poSix. nepejii'^iennx y cnncKy Jiixepaxypn. Hapemxi,
anxop xoxiB 5 H B H C J I O B H X H C B O I O rjra6oKy BAannicxb T . MexcauKijii, I . CjiaByTc;bKO.My i T . A r o 3a p e x e « b n e npoHnxanna nepmoro Bapianxy pyKonncy i
3a "ixni jno6'H3ni 3ayBa>KeimH. HKi AonoMorjin nojiinmnxn BHKJiaA 3 6araxbOx
uorjiHAiB.
.rirrKPATVPA
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72
KompyeHv^ii muny Boponozo Bah hucca BepnyAAi
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HUCAa KAaccoe
udeaAoe
KeadpamuHHUX
noACu,
T p . 4-ro
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[47]
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[49]
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