我们通常将(y=[x])或(y=lfloor x floor)记作关于(x)的取整函数,也称为高斯函数,其意义是不超过x的最大整数
( ext{Lemma 0:})
[lfloor b
floor le b<lfloor b
floor+1
]
( ext{Lemma 0':})
[forall a,b,cin N_+ ,lfloorlfloorfrac{a}{b}
floor/c
floor=lfloorfrac{a}{bc}
floor
]
( exttt{Proof:})
[a=lfloorfrac{a}{b}
floor imes b+r_1=lfloorfrac{a}{bc}
floor imes bc+r_2,r_1in[0, b),r_2in[0, bc),r_2-r_1in(-bc,bc)
]
[lfloorlfloorfrac{a}{b}
floor/c
floor=lfloor frac{a-r_1}{bc}
floor=lfloor lfloorfrac{a}{bc}
floor + frac{r_2-r_1}{bc}
floor=lfloorfrac{a}{bc}
floor
]
( ext{Lemma 1:})
[ain Z,bin R
]
[alelfloor b
floor Leftrightarrow ale b
]
( exttt{Proof:})
[alelfloor b
floor,lfloor b
floorle b Rightarrow ale b
]
[ale b Rightarrow a<lfloor b
floor+1 Leftrightarrow ale lfloor b
floor
]
(整数的离散性:(x,yin Z,x<yLeftrightarrow xle y-1))
( ext{Lemma 2:})
[x,yin Z
]
[xle lfloor frac{n}{y}
floorLeftrightarrow ylelfloor frac{n}{x}
floor
]
( exttt{Proof:})
[ ext{By lemma1:}xlelfloor frac{n}{y}
floorLeftrightarrow xle frac{n}{y} Leftrightarrow ylefrac{n}{x}Leftrightarrow yle lfloor frac{n}{x}
floor
]
( ext{Proposition 3:})
[x,nin Z
]
[xlelfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor
]
( exttt{Proof:})
[ ext{By lemma2: }xlelfloorfrac{n}{lfloorfrac{n}{x}
floor}
floorLeftrightarrowlfloorfrac{n}{x}
floorlelfloorfrac{n}{x}
floor
]
( ext{Theorem 4:})
[xin Z,lfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor}
floor=lfloorfrac{n}{x}
floor
]
( exttt{Proof:})
[ ext{By prosition3: }lfloorfrac{n}{x}
floorlelfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor}
floor--(1),xlelfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor
]
[Rightarrowfrac{n}{x}gefrac{n}{lfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor}gelfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor}
floor
]
[ ext{By lamma1: }lfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor}
floorlelfloorfrac{n}{x}
floor--(2)
]
[(1) ext{ and }(2)Rightarrowlfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor}
floor=lfloorfrac{n}{x}
floor
]
( ext{Corollary 5:})
[yin Z_+,maxleft{xin Z_+|lfloorfrac{n}{x}
floor=lfloorfrac{n}{y}
floor
ight}=lfloorfrac{n}{lfloorfrac{n}{y}
floor}
floor
]
( exttt{Proof:})
[forall xin Z_+ , ext{that } lfloorfrac{n}{x}
floor=lfloorfrac{n}{y}
floor
]
[ ext{By proposition3: }x le lfloorfrac{n}{lfloorfrac{n}{x}
floor}
floor=lfloorfrac{n}{lfloorfrac{n}{y}
floor}
floor
]
[ exttt{original author: 11Dimensions}
]