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].
§1.5GaussBEBZ15
AM1.5.3.CQA0AHxA1CK2[x]+5x?31=0.
C9ARDEBNB51.5.2(3)BIx?1<[x]≤x.AT
x?1<[x]=31?5x2≤x.
CYBJCK
31
7≤x<
33
7,C4D5[x]=4,D5D5ADx=
23
5.square
ASB81.5.4.C3a,bCAAF?CFA9AACNCPADaBRCVbD9CGCFD9AF?CFD9A6CFCV
bracketleftbiga
b
bracketrightbig.
C9ARD6a
b,2b,...,
bracketleftBiga
b
bracketrightBig
b,
ATCXB4BKANA9square
AM1.5.5.B6C8AG350CKBL23CLAPAHCLBGBFAHCLAGAHBL[35023]=15.
C6BJA3B1GaussCRBFB6AHAZCQBQCKBFAHCLAZC7A1CYA3BSCLA9ABA9BLC2A7CTDADEAGALAQCXB1A9
ASB81.5.6.C3x∈R,nCAAF?CFA9AA
bracketleftbigg[x]
n
bracketrightbigg
=
bracketleftBigx
n
bracketrightBig
.
C9ARDEBNB51.5.2(3)BIbracketleftBigx
n
bracketrightBig
≤xn<
bracketleftBigx
n
bracketrightBig
+1.
AGA8
n
bracketleftBigx
n
bracketrightBig
≤x
bracketleftBigx
n
bracketrightBig
+1),n
bracketleftBigx
n
bracketrightBig
≤[x]
bracketleftBigx
n
bracketrightBig
+1),
D5D5ADbracketleftBig
x
n
bracketrightBig
≤[x]n<
bracketleftBigx
n
bracketrightBig
+1.
ACBNB51.5.2(2)BICXB4BKANA9square
BDAP1.5.7.C3m,a,bCAAF?CFA9AA
bracketleftBigm
ab
bracketrightBig
=
bracketleftbigg[m
a]
b
bracketrightbigg
=
bracketleftbigg[m
b]
a
bracketrightbigg
.
DFmA8BGBFAHA7pA8AQAHA9C6mCLAZC7A1CYA3BSAQARAHpCLBMAHBLEp(m).
D9AJ1.5.8.C3nCAD2DJ1D9?CFA7pCACICFA9AA
Ep(n!)=
bracketleftbiggn
p
bracketrightbigg
+
bracketleftbiggn
p2
bracketrightbigg
+···=
∞summationdisplay
r=1
bracketleftbiggn
pr
bracketrightbigg
.
C9ARCFp>nDJA7pB6A6BWBFBWn!,ATC2DJEp(n!)=0.C6DFp≤n.C2DJA7C41CHnBDnAGAH
BSBWARpBFBWCLA8
p,2p,...,
bracketleftbiggn
p
bracketrightbigg
p.
16B2CDCKCLAXBNBQ
AGA8
Ep(n!)=Ep(p·2p····
bracketleftbiggn
p
bracketrightbigg
p)=Ep(p[np]·
bracketleftbiggn
p
bracketrightbigg
!)=
bracketleftbiggn
p
bracketrightbigg
+Ep(
bracketleftbiggn
p
bracketrightbigg
!).
DEBFB41.5.7,ADbracketleftBigg
[np]
p
bracketrightBigg
=
bracketleftbiggn
p2
bracketrightbigg
.
AGA8
Ep(
bracketleftbiggn
p
bracketrightbigg
!)=
bracketleftBigg[n
p]
p
bracketrightBigg
+Ep(
bracketleftBigg[n
p]
p
bracketrightBigg
!)=
bracketleftbiggn
p2
bracketrightbigg
+Ep(
bracketleftbiggn
p2
bracketrightbigg
!).
DFC3AIBFAWCK
Ep(n!)=
bracketleftbiggn
p
bracketrightbigg
+
bracketleftbiggn
p2
bracketrightbigg
+...=
∞summationdisplay
r=1
bracketleftbiggn
pr
bracketrightbigg
.
square
BDAP1.5.9.C3nCAD2DJ1D9?CFA9AA
n!=
productdisplay
p≤n
p
summationtext∞
r=1[
n
pr],
BMAKpCACICFA9
AM1.5.10.CQ30!CLAZC7A1CYA3A9
ADB6C8AG30CLAQAHAD
2,3,5,7,11,13,17,19,23,29.
AGCTAJ1.5.8A6A1B2CQBTBDCKAQARAHAZ30!CLAZC7A1CYA3BSCLBMAHA1B2BL
26,14,7,4,2,2,1,1,1,1.
AGA8
30!=226×314×57×74×112×132×17×19×23×29
DBA830!CLAZC7A1CYA3A9square
AM1.5.11.CQCFC8CLBGBFAHkA1CK10k|199!.
ADC4A1CH10=2×5C1E2(199!)>E5(199!),ADk=E5(199!).ATk=47.square
BDAP1.5.12.C3n=n1+n2+···+ns,BMAKniCAAF?CFA7i=1,2,...,s.AA
n!
n1!n2!···ns!
CA?CFA9
C9ARDEBFB41.4.11,BOCTBHBKD0CZA1AQAHp,AD
Ep(n!)≥Ep(n1!n2!···ns!)=Ep(n1!)+Ep(n2!)+···+Ep(ns!).(1.5.1)
A7A0DAA7DEBNB51.5.2(5),D0CZA1BGBFAHj,AD
bracketleftbiggn
pj
bracketrightbigg
=
bracketleftbiggn
1+n2+···+ns
pj
bracketrightbigg
≥
bracketleftbiggn
1
pj
bracketrightbigg
+
bracketleftbiggn
2
pj
bracketrightbigg
+···+
bracketleftbiggn
s
pj
bracketrightbigg
.
DECTAJ1.5.8,(1.5.1)A3BKANA9square
§1.5GaussBEBZ17
CBAB1.5.13.BFB41.5.12DBA6ABCEBEDGDBBHBKA9A7A0DAA7DFCFADnAGCAADAZC6CLCPBCsAGBF
C9A9CAAZDAAHBDnAGCPC6CHBDsAGBFC9CFBSA7CCBSCO1AGBFC9ADn1AGCPA7CO2AGBFC9ADn2AG
CP...,COsAGBFC9ADnsAGCPA9D1A0D6BHA7DAAGC6CPCLDGDBCCAHDBA8
n!
n1!n2!···ns!,
BDCFCXBHBKARAZA8BFAHA9
BDAP1.5.14.mA6B8D9?CFD9CSAQBJCHm!?CWA9
C9ARC9CXA7BOCTD0mAGAPCZBGBFAHCLCLCQD5CRBHBKA9DFBDmAGBGBFAHBLn+1,n+2,...,n+m,
CCBSnA8BFAHA9CFn=0DJA7CXB4C9CXBKANA9CFnA8BGBFAHDJA7DEBFB41.5.12,
(n+m)!
n!m!=
(n+1)(n+2)···(n+m)
m!
A8BFAHA7C4D5m!|(n+1)(n+2)···(n+m).square
BJB81.5.BHBKC6ASAHB5A9
1.CYDGBMx+4{x}=2[x].
2.CYDGBMx2?2[x]?5=0.
3.DFx>0CK
[x]
{x}=
x
[x].CQx.
4.CQDGBM[1.9x]+[9.2y]=37CLBGBFAHCYA9
5.DFx,y∈R.BHBK[x?y]≤[x]?[y].
6.DFx∈R.BHBK[x]+[x+12]=[2x].
7.DFx,y∈R.BHBK[2x]+[2y]≥[x]+[y]+[x+y].
8.CQ20!CLAZC7A1CYA3A9
9.CQ78!CLBPBNAPCZCLATCLAGAHA9
10.DFnA8BGBFAHA9BHBKn!(n?1)!|(2n?1)!.
11.DFpA8AQAHA7αA8BGBFAHA9BHBKB6C8AGpαCKAIpαBKAQCLBGBFAHADpα?pα?1AGA9
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