4B2CDCKCLAXBNBQ
§1.2CED4A3C1B2?C3CCBND0DG
ASCWCKANCFC8ANARAHCLAABZC1CEAUCLDECKBVASA7A0A7BCDJBDAGCQCZA1AQBFAHCFC8ANARAHCL
CFBVASDGDB–B5C5CEBWDBA9C7A3B1AQAGBFAHCLCFC8ANARAHA9
D9BU1.2.1.DFa,b,c∈Z.BJcA8a,bCLANARAHA7D6c|aCKc|b.D6a,bB6CUBLATA7B2BJa,bCLAN
ARAHBSBJCFC8BCBLa,bCLCFC8ANARAHA9AYCT0AI0CLCFC8ANARAHBL0.AQBFAHa,bCLCFC8ANAR
AHC6BL(a,b).D6(a,b)=1,B2BJa,bBKAQA9ACCTA2D1A0A2BTA7AQB6CUBLATCLBFAHCLCFC8ANARAH
ATBLBGBFAHA9
AM1.2.2.ABCFC8ANARAHCTA2CQ24BC45CLCFC8ANARAHA9
ADD1A0D6BHA724CLARAHAD
±1,±2,±3,±4,±6,±8,±12,±24,
D545CLARAHAD
±1,±3,±5,±9,±15,±45.
AT24BC45CLANARAHAD±1,±3,C4D5CFC8ANARAHA83,C2(24,45)=3.square
AM1.2.3.DFmA8BGCDAHA7nA8BGBFAHA9B2(2m?1,2n+1)=1.
C9ARDFd=(2m?1,2n+1).B2C5AZs,t∈ZA1CK2m=ds+1,2n=dt?1.AGA8(ds+1)n=
(dt?1)m.C4A1CHmA8CDAHA7ACC0D1D7CGA3CTAJA7C5AZu,v∈ZA1CKdu+1=dv?1.BDB1BK
d|2.CE2m?1BC2n+1A0BLCDAHA7ATd=1.square
C6BJCLCXB4A8C9CXCLA9
ASB81.2.4.C3a,b∈Z.AA
(1)a,bA1|a|,|b|DHCSBVD2CQD9A8A6CFA9
(2)(a,b)=(|a|,|b|)=(b,a).
(3)(a,1)=1,(a,0)=|a|.
(4)BZa|b,AA(a,b)=|a|.
C4DAAGBNB51.2.4A6DJA2BTA7A3B1AQBFAHCLCFC8ANARAHCFBTA6AZCXBLA3B1AQBGBFAHCLCFC8
ANARAHA9C6BJD1DCCQCZA1D7BFAHCLCFC8ANARAHCLDGDBA9C7A2C6BJCLBVASA7A0A9
ASB81.2.5.C3a,b,q,r∈ZBRa=qb+r.AA(a,b)=(b,r).
C9ARD6u|a,u|b,B2u|bCKu|a?qb=r.DDBJA7D6u|b,u|r,B2u|bCKu|qb+r=a.BDB1BKa,bAI
b,rCLANARAHBHCUCEBCA7C4D5CFC8ANARAHCEBCA7C2(a,b)=(b,r).square
D9AJ1.2.6.A5ABAND2CWDHA6C3a,b∈Z,b>0.DIC3
?a=q1b+r1(0
?b=q2r1+r2(0
?······
?rn?2=qnrn?1+rn(0
§1.2CPAZBBCHBZ?CJCNC9AXB65
?rn?1=qn+1rn+rn+1(rn+1=0).
AT
P0=1,P1=q1,Pk=qkPk?1+Pk?2,
Q0=0,Q1=1,Qk=qkQk?1+Qk?2,
k=2,3,...,n.
AADDCZA7B4CRB7AB
(1)(a,b)=rn.
(2)ACDJBXDEk=1,2,3,...,n,DH
parenleftBigg
q11
10
parenrightBiggparenleftBigg
q21
10
parenrightBigg
···
parenleftBigg
qk1
10
parenrightBigg
=
parenleftBigg
PkPk?1
QkQk?1
parenrightBigg
.
(3)ACDJBXDEk=1,2,3,...,n,DHrk=a[(?1)k?1Qk]+b[(?1)kPk].
C9AR(1)DEBNB51.2.4BC1.2.5,AD
(a,b)=(b,r1)=(r1,r2)=···=(rn?1,rn)=(rn,rn+1)=(rn,0)=rn.
(2)CFk=1DJA7CXB4C9CXBKANA9DFCXB4D0k=t?1BKANA7C2
parenleftBigg
q11
10
parenrightBiggparenleftBigg
q21
10
parenrightBigg
···
parenleftBigg
qt?11
10
parenrightBigg
=
parenleftBigg
Pt?1Pt?2
Qt?1Qt?2
parenrightBigg
.
B2CFk=tDJA7ADparenleftBigg
q11
10
parenrightBiggparenleftBigg
q21
10
parenrightBigg
···
parenleftBigg
qt?11
10
parenrightBiggparenleftBigg
qt1
10
parenrightBigg
=
parenleftBigg
Pt?1Pt?2
Qt?1Qt?2
parenrightBiggparenleftBigg
qt1
10
parenrightBigg
=
parenleftBigg
PtPt?1
QtQt?1
parenrightBigg
.
ACAZBTDBAOAJA7CXB4BKANA9
(3)CFk=1DJA7CXB4C9CXBKANA9DFCXB4D0k
rt?1=a[(?1)t?2Qt?1]+b[(?1)t?1Pt?1],rt?2=a[(?1)t?3Qt?2]+b[(?1)t?2Pt?2].
AT
rt=rt?2?qtrt?1=a[(?1)t?1Qt]+b[(?1)tPt].
ACAZBTDBAOAJA7CXB4BKANA9square
DECTAJ1.2.6BJBAA9(1)BC(3),C6AGCXB4A8C9CXCLA9
BDAP1.2.7.C3a,b∈Z.AAD0A8s,t∈ZC7D8(a,b)=as+bt.
AM1.2.8.CQs,t∈ZA1CK(2346,1081)=2346s+1081t.
6B2CDCKCLAXBNBQ
ADC4A1CH
2346=2×1081+184,1081=5×184+161,184=1×161+23,161=7×23+0,
DECTAJ1.2.6BJBHBKB2BMA7AD(2346,1081)=23CK
23=2346[(?1)3?1Q3]+1081[(?1)3P3],
CCBSparenleftBigg
P3P2
Q3Q2
parenrightBigg
=
parenleftBigg
21
10
parenrightBiggparenleftBigg
51
10
parenrightBiggparenleftBigg
11
10
parenrightBigg
=
parenleftBigg
1311
65
parenrightBigg
.
AGA8CRs=6,t=?13C2A6A9square
ASB81.2.9.C3a,b,c∈Z.
(1)(a,b)=1D5BRAZD5D0A8s,t∈ZC7D8as+bt=1.
(2)BZc|a,c|b,AAc|(a,b).
(3)BZ(a,c)=1,AA(ab,c)=(b,c).
(4)BZ(a,c)=1,c|ab,AAc|b.
(5)BZ(a,c)=(b,c)=1,AA(ab,c)=1.
(6)BZ(a,b)=1,a|c,b|c,AAab|c.
C9AR(1)ATDACSACBFB41.2.7ANCKA9DDBJA7D6x|a,x|b,B2x|as+bt=1.BDB1BKxBOBWBL±1,AG
A8a,bCLANARAHBOAD±1,AT(a,b)=1.
(2)DEBFB41.2.7BIC5AZs,t∈ZA1CK(a,b)=as+bt.AGA8ACc|a,c|bA6BIc|as+bt=(a,b).
(3)C9CXA7b,cCLANARAHA0BLab,cCLANARAHA9AWDEDGBJA7AC(a,c)=1C1BAA9(1)BIC5AZ
s,t∈ZA1CKas+ct=1.AGA8b=abs+bct.BDCGBQab,cCLANARAHA0BLb,cCLANARAHA9ATb,cBC
ab,cADBHCUCEBCCLANARAHA7C4D5ADCEBCCLCFC8ANARAHA7C2(ab,c)=(b,c).
(4)AC(a,c)=1C1BAA9(1)BIC5AZs,t∈ZA1CKas+ct=1.AGA8ACc|abBIc|abs+bct=b.
(5)AC(a,c)=(b,c)=1C1BAA9(1)BIC5AZs,t,u,v∈ZA1CKas+ct=1=bu+cv.AGA8
ab(su)+c(asv+but+ctv)=1.
DEBAA9(1)BI(ab,c)=1.
(6)ACBAA9(1)C1(a,b)=1BIC5AZs,t∈ZA1CKas+bt=1.ACa|c,b|cA6CKab|bc,ab|ac.AT
ab|acs+bct=c.square
ASB81.2.10.C3a,bCACNBVCVBBD9?CFA7mCAAF?CFA9AA(am,bm)=(a,b)m.
C9ARC6d=(am,bm).ACBNB51.2.9(2)C1(a,b)m|am,(a,b)m|bmBI(a,b)m|d.AWDEDGBJA7DEBF
B41.2.7BIC5AZs,t∈ZA1CK
(a,b)m=ams+bmt.
ACAGd|am,d|bm,ATd|(a,b)m.BDB1BK(am,bm)=(a,b)m.square
BDAP1.2.11.C3a,bCACNBVCVBBD9?CFA7mCAAF?CFA9BZm|a,m|b,AA
(am,bm)=(a,b)m.
§1.3CPCBBBATBZ7
CNCMD9A7
(a(a,b),b(a,b))=1.
AIAPAGAQAGBFAHCLCLCQA7A6DJA3B1D3AGBFAHCLCFC8ANARAHA9
D9BU1.2.12.DFa1,a2,...,ak∈Z(k≥2),d∈Z.BJdA8a1,a2,...,akCLA8A6CF,D6
d|ai,i=1,2,...,k.
D0B6CUBLATCLBFAHa1,a2,...,ak,BJAYBGCLANARAHBSBJCFC8BCBLa1,a2,...,akCLCFC8ANARAHA7C6
CI
(a1,a2,...,ak).
D0AGa1,a2,...,akCUBL0CLCLAAA7AYCTCCCFC8ANARAHBL0.
D6(a1,a2,...,ak)=1,B2BJa1,a2,...,akBKAQA9D6a1,a2,...,akBSCZA1AQAGAHA0BKAQA7B2
BJa1,a2,...,akAQAQBKAQA9C9CXA7AQAQBKAQATBKAQA7CEDDBJBOATA9
C6AGBNB5CMCQD3AGBFAHCLCFC8ANARAHCLBUB5AZCXBLCQAQAGBFAHCLCFC8ANARAHCLBUB5A9
ASB81.2.13.C3ai∈Z,i=1,2,...,n.AT
d1=a1,(dj?1,aj)=dj,j=2,3,...,n.
AA(a1,a2,...,an)=dnBRD0A8s1,s2,...,sn∈ZC7D8
a1s1+a2s2+···+ansn=dn.
BJB81.2.BHBKC6ASAHB5A9
1.CQ(51425,13310)C1BFAHs,tA1CK(51425,13310)=51425s+13310t.
2.BHBKBNB51.2.13.(B4A4ABD0nABAHD1AZBTDBB4ALABBFB41.2.7)
3.CQ(353430,530135,165186).
4.DFanxn+an?1xn?1+···+a1x+a0A8BFC5AHD3CGA3A9AFDFu,v∈Z,(u,v)=1CK
v
uA8A9
D3CGA3CLAJA9BHBKv|a0,u|an.
5.DFa,bA8BFAHA7mA8BGBFAHCKm|a,m|b.AFDFC5AZBFAHs,tA1CKm=as+bt.BHBK(a,b)=m.
6.DFa,b,nA0BLBGBFAHA9BHBK(an,bn)=(a,b)n.
§1.3CEBPA3COB2
ASCWB2B4BFAHCLCFCIANAPAHA9AICFC8ANARAHCLCQDED9A7C7A3B1AQAGBFAHCLCLAAA9
D9BU1.3.1.DFa,bA0BLDJATBFAHA9BJd∈ZBLa,bCLA8CGCF,D6a|d,b|d.BJa,bCLBGANAPAHBJ
CFCIBCBLa,bCLCFCIANAPAHA7C6CI[a,b].
C6BJCLBNB5A8C9CXCLA9
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