§1.4C2BYBIAUB3BN9
ASB81.3.8.C3a1,a2,...,akCAA1BB?CFA9AT
m1=a1,[mj?1,aj]=mj,j=2,3,...,k.
AA[a1,a2,...,ak]=mk.
BJB81.3.BHBKC6ASAHB5A9
1.C5AT[391,493].
2.CQ[209,665,4025]B4ALBK
[209,665,4025]negationslash=209×665×4025(209,665,4025).
3.D2ADCCDIAQAGBOB3BKCEC8CTA7CCB3AD437BOA7DIB3AD323BOA9CCCLBQDEBOAIDICLBQDEBOCE
CTBZBICHAYBKCECTBZA7BPDBAHDAC5C3BX?
4.DFa,b,cA8AQAQB6BCCLBGBFAHCKAQAQBKAQA9AFDFD7AGAHBSCZBDAQAGBJBCA0A6ARCOD7AGBFBWA9
CQa,b,c.
5.DFBQD7BQAHA8CCAHBQAHB7BJBCCL12APA9CQBDAGD7BQAHA9
6.DFa,bA8BGBFAHCK[a,b]=(a,b).BHBKa=b.
7.DFa,bBLBGBFAHA9BHBK(a+b)[a,b]=a[b,a+b],(B4A4ABABBFB41.3.4BCBNB51.2.5.)
8.BHBKBNB51.3.8.
§1.4B6B1AACPDBAL
ASCWD1DCAQAHA7BEAHCMAABZA7AZC2BVBXDACKANATAFBVASCTAJB4A3B1BDDECTAJCLDECKA9ABA9
ABC7AIBTAQAHCLAABZA9
D9BU1.4.1.DFa∈Z,a>1.BJaA8CICF,D6CCBGARAHBOAD1C1CCASDGA9A2B2A7BJCCBLAICF.
CBAB1.4.2.AQAHDBBJBLBRAHBUB6A6ARAHADAQAHA0BLBGBFAHADCUB6BGBFAHA1BL1,AQAHBCBEAHD7
AIA9
AM1.4.3.DFnA8BGBFAHA9B2
(n+1)!+2,(n+1)!+3,···,(n+1)!+n+1
A8APCZCLnAGBEAHA9
D9AJ1.4.4.AICFaD9CW1DDCRD9ASD4AFA6CFCACICFBRCNCPAD
√a.
C9ARDFaCLBW1DJBGCLCFCIBGARAHA8pCKa=pq.D6pBLBEAHA7B2C5AZp1,p2∈ZA1CKp=p1p2
CK1
A8BEAHA7ADqnegationslash=1.ACpCLCFCICSBIp≤q.ATa=pq≥p2.C2p≤√a.square
BDAP1.4.5.C3aCAD2DJ1D9?CFA9BZCNCPAD
√a
D9CICFB2CNBJ?CWa,AAaCACICFA9
10B2CDCKCLAXBNBQ
AM1.4.6.B9BTB6BGB2100CLCUB9AQAHA9
ADC4A1CHB6BGB2
√100=10
CLAQAHBL2,3,5,7,DEBFB41.4.5,B6BGB2100CLBEAHATBWAR2,3,5,7
BSBJDEBFBWA9AGA82,3,5,7AYCBDA2CH100BJCFB6BWAR2,3,5,7BSCZBDDEAGBFBWCLAHDBA8AXCQ
CLCUB9AQAHA9BDCKAQAHA8AB2,3,5,7,11,13,17,19,23,29,31,37,41,43,
47,53,59,61,67,71,73,79,83,89,97.square
D9AJ1.4.7.(Euclid)CICFDHCXD1A6A9
C9ARDFp1,p2,...,pnA8AXADAQAHA9B2
a=p1p2···pn+1
A8BEAHA9DECTAJ1.4.4BIaCLBW1DJBGCLCFCIBGARAHpA8AQAHA9BXDHDFp=p1.B2
p1=p|a=p1p2···pn+1.
BDCGBQp1|1,BBD2A9square
ASB81.4.8.C3a,b∈Z,pCACICFA9AA(a,p)=1AOp|a.BCCRA7BZp|ab,AApAIC2?CWa,bAKD9
DCA6A9
C9ARAC(a,p)|pC1pA8AQAHBI(a,p)=1BU(a,p)=p.CF(a,p)=pDJA7AC(a,p)|aBIp|a.ATA9
BNB5BJCIDECXB4BKANA9DFp|ab.D6p?a,B2ATAD(p,a)=1.DEBNB51.2.9(4)BIp|b.square
BDAP1.4.9.BZai∈Z,i=1,2,...,n,pCACICFBRp|a1a2···an,AApAIC2?CWa1,a2,...,anAKD9
DCA6A9
D9AJ1.4.10.(CJCDAPCIDCB6)BXAHD2DJ1D9?CFaB2B3CUDCCLC9CV
a=p1p2···pn,
BMAKp1,p2,...,pnCACICFBRp1≤p2≤···≤pn.
C9ARD0aABAHD1AZBTDBA9CFa=2DJA7CXB4C9CXBKANA9CADFCXB4D0C8AG1CIAGaCLDECJBFAHBK
ANA9C6BHCXB4D0BFAHaDBBKANA9D6aA8AQAHA7B2CXB4C9CXBKANADD6aA8BEAHA7B2a=bc,CCBSb,c
A0BLC8AG1CIAGaCLBFAHA9ACAZBTCDDFA7b,cA0A6DJCLBKDECKAQAHCLBLBXA9AGA8A7aDBA6DJCL
BKD6ABAQAHCLBLBXA9CMBDCKAQAHAGACCICHC8C4ASC2A6CKa=p1p2···pn,CCBSp1,p2,...,pnA8AQ
AHCKp1≤p2≤···≤pn.D6BOADa=q1q2···qm,CCBSq1,q2,...,qmA8AQAHCKq1≤q2≤···≤qm,
B2
p1p2···pn=q1q2···qm.
BDALBKp1|q1q2···qm.ACBFB41.4.9BIC5AZqjA1CKp1|qj.D0C3CLA7C5AZpiA1CKq1|pi.ACAG
p1,q1,pi,qjA0BLAQAHA7ATp1=qj,q1=pi.C4A1CHp1≤pi=q1≤qj=p1,ADp1=q1,C4
D5p2···pn=q2···qm.AIAPDAAGB2B4A6CKp2=q2.DFC3D5CRC6CTAWBIn=mCKpi=qi,i=
1,2,...,n.square
BDAP1.4.11.BXAHD2DJ1D9?CFaB2B3CUDCCLC9CV
a=pα11pα22···pαnn,
BMAKp1,p2,...,pnCACICFA7αiCAAF?CFBRp1
D9CKAOA2AYC8A9CXC6A7p1,p2,...,pnCAaD9BVCOCIA6CFA9
§1.4C2BYBIAUB3BN11
AM1.4.12.BGBFAH51480CLAZC7A1CYA3A851480=23×32×5×11×13.
AM1.4.13.4n+3CPAQAHADBXCND3AGA9
C9ARDFp1=3,p2,...,ptA8AXAD4n+3CPAQAHA9B2a=4p2···pt+3A8BEAHA9BWBGCZD5A7a
BX4n+3CPAQARAHA9A2B2A7DFqA8aCL4n+3CPAQARAHA9D6q=3,B2ACq|aBI3|4p2···pt.
CE(3,4)=1,C4D53|p2···pt.DEBFB41.4.9BI3BFBWBQAGpi,(i≥2).BDCGBQpi=3,BBD2A9D6
q=pj,j≥2.B2ACq|aBIq|3,D5D5ADq=pj=3,BBD2A9AGA8A7DECTAJ1.4.10BIaBLD6AB4n+1
CPAQAHCLBLBXA7CE4n+1CPAQAHCLBLBXBL4n+1CPAHA7AIaA84n+3CPAHBBD2A9square
ASB81.4.14.C3D2DJ1D9?CFaD9CKAOA2AYC8CV
a=pα11pα22···pαnn.
AAaD9CLDHAFA6CFCV
F={pε11pε22···pεnn|0≤εi≤αi,i=1,2,...,n}.
C9ARC9CXBZBEFBSCLAMAQA0BLaCLBGARAHA9DDBJA7DFd|a.D6d=1,B2C9CXADd∈F.C6DF
dnegationslash=1.C7CWCTdCLAQARAHA9DFpA8dCLAQARAHA9B2pDBA8aCLAQARAHA7C4D5pDBA8p1,p2,...,pn
BSCLBQDEAGA9DEBFB41.4.11BIdADD3C6B1C6A3d=pε11pε22···pεnn.ACd|aBI
0≤εi≤αi,i=1,2,...,n.
BDB1BKd∈F.square
BDAP1.4.15.C3a,bCAD2DJ1D9?CFA7
a=pα11pα22···pαnn,b=pβ11pβ22···pβnn,
BMAKpiCACICFA7αi,βjCAA1A4?CFA7i,j=1,2,...,nBRp1
(a,b)=pγ11pγ22···pγnn,[a,b]=pδ11pδ22···pδnn,
BMAK
γi=min{αi,βi},δi=max{αi,βi},i=1,2,...,n.
AM1.4.16.AC
51480=23×32×5×70×11×13
BC
41580=22×33×5×7×11×130
BI
(51480,41580)=22×32×5×70×11×130=1980,
[51480,41580]=23×33×5×7×11×13=1081080.
CTDABMBTCLA8A7BFB41.4.15AXBDAGCLCQCFC8ANARAHBCCFCIANAPAHCLDGDBB6BWCBB7B5C5CEBW
DBA7A6BLCQDEAGAHCLAZC7A1CYA3BPD2BEADBDDECLAMDBA9
BDAP1.4.17.C3D2DJ1D9?CFaD9CKAOA2AYC8CVa=pα11pα22···pαnn.AAaD9CLDHAFA6CFD9AGCV
S(a)=p
α1+1
1?1
p1?1·
pα2+12?1
p2?1···
pαn+1n?1
pn?1;
aD9AFA6CFD9A6CFCV
τ(a)=(α1+1)(α2+1)···(αn+1).
12B2CDCKCLAXBNBQ
BDAP1.4.18.C3a,bCAAF?CFBR(a,b)=1.AA
τ(ab)=τ(a)τ(b),S(ab)=S(a)S(b).
AM1.4.19.ACAG28=22×7,AT
τ(28)=(2+1)(1+1)=6,S(28)=2
3?1
2?1
72?1
7?1=56.
CIBLDABJCXB1CLAUABA7D1DCMersenneAQAHBCBHCUAHB4CSA4AZBGBJCFCLD3BHAOC5A9ABC7CT
A2MersenneAQAHA9BLC2A7C7AIBTDJC6A7A0A9
ASB81.4.20.BZa,nCAAF?CFA7n>1BRan?1CVCICFA7AAa=2BRnCACICFA9
C9ARD6a>2,B2(a?1)|an?1CK1
n=kl,1
D9BU1.4.21.BJCQD32p?1CLAQAHBLMersenneAQAHA9ACDAAGBNB51.4.20BIA7D62p?1BLMersenne
AQAHA7B2pATBLAQAHA9BJBGBFAHnBLBHCUAHA7D6S(n)=2n,AMD36,28A0BLBHCUAHA9
AUAGMersenneAQAHBCBHCUAHA7BWBGADDJC6BVDACXB1A9
D9AJ1.4.22(Euclid-Euler).C3q=2p?1CVMersenneCICFA9AA
1
2q(q+1)=2
p?1(2p?1)=2p?1q
CACSBVCFA9DIAGA7BXAHBKCSBVCFB2B3BYCXAAA9A9
C9ARDEBFB41.4.17BI
S(12q(q+1))=S(2p?1q)=2
p?1
2?1
q2?1
q?1=q(q+1),
AT
1
2q(q+1)BLBHCUAHA9DDBJA7DFaA8DEC3BHCUAHA9D6a=2
n,n≥1,
B2
2n+1=2a=S(a)=2
n+1?1
2?1=2
n+1?1,
BBD2A9ATA6C6a=2n?1u,u>1,n≥2,2?u.DEBFB41.4.18C1aA8BHCUAHBI
2nu=2a=S(a)=2
n?1
2?1S(u).
AT
S(u)=2
nu
2n?1=u+
u
2n?1.
CEuC1
u
2n?1CUBLuBJBGARAHA7D5S(u)BLuCLAXADBGARAHBJBCA7BDB1BKuBOADAQAGBGARAHA7
ATuA8AQAHCK
u
2n?1=1.AGA8u=2
n?1
A8MersenneAQAHA7C4D5nA8AQAHCKa=2n?1u.square
ANAMCI6A6C9CLPythagorasA8CFB0D4D9BHCUAHCLCYA7AYDHD7BICI6BC28A8BHCUAHA9CTAJ
1.4.22CLBGBJB9A1A8EuclidAZA3C3BDAOASA4BSBHBKCLADD5CCDDBJB9A1B2A8EulerBHBKCLA9AWDE
DGBJA7DBB0CYMersenneABC7D0CQD32
p?1
CLAQAHB6A1C5BDD4D9A9BLC9BZAYA7DBAHCQD32p?1
CLAQAHBJBLMersenneAQAHA9ACAGMersenneAQAHBCBHCUAHCLBVDAAOC5DJC1AZCCAZAVAJCLAXDFA9
ABA7CGAIBYAGDEBKC1A7C8BWD3AHD1CABCBXAHCLAHD1AFBABCD0AZD5CRB1D9A9CEMersenneAQAHAZ
BGBFAHBSCLA1B7A5BEC2ADA7CVBNCH2014BY2ASA7AOD9CA48AGMersenneAQAHA9DECTAJ1.4.22,
DBDBADAR48AGBHCUAHA7BDCKBHCUAHA0BLC3AHA9A8A2C5AZCDBHCUAHA8AHB4BSDEAGBOCYDICLBUB5A9
ASCWCFBID1DCADCSCLFermatAHA7ALABAZA6DJCKCH“AQAHBXCCAG”BDDEBVDACXB4CLDEAGCM
BHBKA9
§1.4C2BYBIAUB3BN13
D9BU1.4.23.DFmA8DJA8BFAHA9BJFm=22m+1BLFermatAHA9CIC3AGFermatAHBL
F0=3,F1=5,F2=17,F3=257,F4=65537,F5=641×6700417,
CCBSA7CFBIDEAGCMA3A8EulerAG1732BYBMBTCLA9
BZFermatAHBJCFADDJC6BVDAAUC5A9
D9AJ1.4.24.ACDJBXDEAF?CFn,DH
n?1productdisplay
k=0
Fk=Fn?2.
C9ARD0nABAHD1AZBTDBA9ACAGF0=3=5?2=F1?2,ATCFn=1DJCXB4BKANA9DFCXB4D0
n=tDJBKANA7C2
t?1productdisplay
k=0
Fk=Ft?2.
B2CFn=t+1DJA7AD
n?1productdisplay
k=0
Fk=
tproductdisplay
k=0
Fk=
t?1productdisplay
k=0
Fk·Ft=(Ft?2)Ft
=F2t?2Ft=(22t+1)2?2(22t+1)=Ft+1?2=Fn?2.
BDALBKCXB4D0n=t+1CLCLCQDBBKANA9ACAZBTDBAOAJA7BNB5D0CZA1BGBFAHnCVBKANA9square
BDAP1.4.25.CNCQD9B9A6FermatCFALCIA7CZDFCICFDHCXD1A6A9
C9ARDFm,nA8BGBFAHCKm
Fn=F0F1···Fm?1FmFm+1···Fn?1+2=(F0F1···Fm?1Fm+1···Fn?1)Fm+2.
DEBNB51.2.5C1FermatAHA0BLCDAHBDDEA7A0A7AD(Fn,Fm)=(Fm,2)=1.ATB6BCCLAQAGFermat
AHBKAQA9
DFAQAHBOADADCCD3AGA9DEATAFBVASCTAJA7BFAGFermatAHCVADAQARAHA9ACAGFermatAHADBX
CNAGA7C4D5ATADAQAGFermatAHADCEBCCLAQARAHA7BDAIB6BCCLAQAGFermatAHBKAQBBD2A9ATAQAH
ADBXCCAGA9square
CTDAB3B2BMBTCLA8A7FermatAHA6CCC3BDDACLB3ACA9ABCKCHCYBGCLAUC4A9BDA8A6BLA7Gauss
B3BHBKA7D6FnA8AQAHA7B2BGFnAVCQA6BPAYCIBEA9
BJB81.4.BHBKC6ASAHB5A9
1.B9BTB6BGB2300CLCUB9AQAHA9
2.CQ16500BC1452990CLAZC7A1CYA3B4CQ(16500,1452990)BC[16500,1452990].
3.DHBIx2+px+q=0CLAQAJBLBGBFAHCKp+q=28.CQA9DGBMCLAQAGAJA9
4.DFp,qA8AQAHA7α,βA8x2?px+q=0CLAQAGBGBFAHCYA9CQα+β,p,qCLBLA9
5.DFAQAHp≤1000CK2p+1=am,CCBSa,m∈Z,a>0,m>1.CQpBCmCLBLA9
14B2CDCKCLAXBNBQ
6.D6AQAHpBFBWBQC8DGAHa,B2p2DBBFBWa.
7.DFl,m,n∈N,lA8AQAHCKl2+m2=n2.BHBK2(l+m+1)A8C8DGAHA9
8.DFpA8AQAHA9BHBKp|Cip,i=1,2,...,p?1.
9.BHBK
√p
A8BXAJAHA7CCBSpA8AQAHA9
10.BHBK6n+5CPAQAHADBXCND3AGA9
11.DFa,b,cA8BGBFAHA9BHBK(a,[b,c])=[(a,b),(a,c)].(B4A4ABABBFB41.4.15.)
12.BHBKlog102,log73,log2115A0BLBXAJAHA9
13.DFnA8BGBFAHCKnCLBWnDJBGCLBGARAHBJBXCMAGn.BHBKnDABCBLDEAQAHBJANDGA7DABCBL
D7B6BCAQAHBJBXA9
14.DFpA8AQAHCKp4CLCUB9BGARAHBJBCBLC8DGAHA9CQp.
15.CQ30!CLBGARAHCLAGAHA9
16.CQ51480CLBGARAHBJBCA9
17.DF2n+1(n>1)A8AQAHA9BHBKnA82CLDGBIA9
18.DFnA8BGBFAHA9BHBKnA8AQAHCFCKD4CFS(n)=n+1.
19.DFnA8BGBFAHA9BHBKnA8BHCUAHCFCKD4CF
summationdisplay
d|n
1
d=2.
20.DFnA8BGBFAHA9BHBKnA8C8DGAHCFCKD4CFnCLBGARAHAGAHBLCDAHA9
§1.5GaussA6B2
ASCWD1DCGaussCRBFB6AH[x]C1CCAZCQBGBFAHAZC7A1CYA3CMDGBJCLA9ABA9C7A2C6BJCLCTA2A9
D9BU1.5.1.DFx∈R.DJ[x]B1A4B6BGB2xCLCFC8BFAHB4BJCCBLxCL?CFCOA2.BJ{x}=x?[x]
BLxCLD4CFCOA2.AMD3A7
[pi]=3,[?pi]=?4,
bracketleftbigg2
3
bracketrightbigg
=0,
bracketleftbigg
?35
bracketrightbigg
=?1,{32}={?32}=12.
D1A0A2BTA7D0CZA1x∈R,AD0≤{x}<1.
C6BJCLBNB5A8D1A0D6BHCLA9
ASB81.5.2.C3x,y∈R,m∈Z.AA
(1)x≤y=?[x]≤[y].
(2)m≤x
(3)x?1<[x]≤x<[x]+1.
(4)[m+x]=m+[x].
(5)[x]+[y]≤[x+y].
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