下取整函数
上取整函数
在数学和计算机科学中,取整函数是一类将实数映射到相近的整数的函数。[1]
常用的取整函数有两个,分别是下取整函数(英語:floor function)和上取整函数(ceiling function)。
下取整函数即為取底符號,在数学中一般记作
或者
或者
,在计算机科学中一般记作floor(x),表示不超过x的整数中最大的一个。
![{\displaystyle [x]=\max \,\{n\in \mathbb {Z} \mid n\leq x\}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84yqe0ajs3nDBEoDJAaDoNaAi5yta1zjK1ntmQaqw1o2o0otGNaAnF)
举例来说,
,
,
,
。对于非负的实数,其下取整函数的值一般叫做它的整数部分或取整部分。而
叫做x的小数部分。每个分数都可以表示成其整数部分与一个真分数的和,而实数的整数部分和小数部分是与此概念相应的拓延。
下取整函数的符号用方括号表示(
),称作高斯符号,首次出現是在卡爾·弗里德里希·高斯的數學著作《算术研究》。
上取整函数即為取頂符號在数学中一般记作
,在计算机科学中一般记作ceil(x),表示不小于x的整数中最小的一个。
![{\displaystyle \lceil x\rceil =\min\{n\in \mathbb {Z} \mid x\leq n\}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qa2dCzNdEoAvFngnCzNs5yjFCztoNoAnDatCOyjhCoto1otwOygdA)
举例来说,
,
,
,
。
计算机中的上取整函数和下取整函数的命名来自于英文的ceiling(天花板)和floor(地板),1962年由肯尼斯·艾佛森于《A Programming Language》引入。[2]
对于高斯符號,有如下性质。
- 按定义:
当且仅当x为整数时取等号。
- 设x和n为正整数,则:
![{\displaystyle \left[{\frac {n}{x}}\right]\geq {\frac {n}{x}}-{\frac {x-1}{x}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nqi4agnEz2hAatzEztzBytK2aAs2yto2o2zEa2i3zjJEnDJEztdD)
- 当n为正整数时,有:
其中
表示
除以
的餘數。
- 对任意的整数k和任意实数x,
![{\displaystyle [{k+x}]=k+[x].}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80nDwNaNm1o2iNnDs5ataOzDzBa2s4aAo4nti3agnDzDGOaDlFytdF)
- 一般的數值修約規則可以表述为将x映射到floor(x + 0.5);
- 高斯符號不是连续函数,但是上半连续的。作为一个分段的常数函数,在其导数有定义的地方,高斯符號导数为零。
- 设x为一个实数,n为整数,则由定义,n ≤ x当且仅当n ≤ floor(x)。
- 當x是正數時,有:
![{\displaystyle \left\lbrack 2x\right\rbrack -2\left\lbrack x\right\rbrack \leqslant 1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoqzBoNKNyqnBoNhEzghDzDeQoDsOygzBoNCNagdCajlCo2wPnDC0)
- 用高斯符號可以写出若干个素数公式,但没有什么实际价值,見§ 質數公式。
- 对于非整数的x,高斯符號有如下的傅里叶级数展开:
![{\displaystyle [x]=x-{\frac {1}{2}}+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zNBEntK2oNG1oqaQyqi4aAwPzjJEzqiQztG3zDdDoDmQatdDoAeQ)
- 根据Beatty定理,每个正无理数都可以通过高斯符號制造出一个整数集的分划。
- 最后,对于每个正整数k,其在 p 进制下的表示有
个数位。
由上下取整函數的定義,可見
![{\displaystyle \lfloor x\rfloor \leq \lceil x\rceil ,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzNrFoqeOzqeOngzEngnBzgzBztBAaNePnji3ztw4otaQoNa4yjFA)
等號當且僅當
為整數,即
![{\displaystyle \lceil x\rceil -\lfloor x\rfloor ={\begin{cases}0,&{\text{ 若 }}\ x\in \mathbb {Z} ,\\1,&{\text{ 若 }}\ x\not \in \mathbb {Z} .\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81otC5yqs1oAwPytBAoqnDotK1nAaPagw5zAiNoDm2yjvFzjlDnDw0)
實際上,上取整與下取整函數作用於整數
,效果等同恆等函數:
![{\displaystyle \lfloor n\rfloor =\lceil n\rceil =n.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81otG4otvEoNm0nqw4ztrBaqs3ateNatGPagiNo2i4zDo2aghFnjw5)
自變量加負號,相當於將上取整與下取整互換,外面再加負號,即:
![{\displaystyle {\begin{aligned}\lfloor x\rfloor +\lceil -x\rceil &=0,\\-\lfloor x\rfloor &=\lceil -x\rceil ,\\-\lceil x\rceil &=\lfloor -x\rfloor .\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Cytw1ygsQagdEaNFDzjm4nDrBytvCaNG5njJBzgvEa2hBzjCPztK3)
且:
![{\displaystyle \lfloor x\rfloor +\lfloor -x\rfloor ={\begin{cases}0,&{\text{ 若 }}\ x\in \mathbb {Z} ,\\-1,&{\text{ 若 }}\ x\not \in \mathbb {Z} ,\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DoNhBnjCOntlDygs5zDiOotm4zqe1aNrEo2a1oti5agw4yqvEo2zB)
![{\displaystyle \lceil x\rceil +\lceil -x\rceil ={\begin{cases}0,&{\text{ 若 }}\ x\in \mathbb {Z} ,\\1,&{\text{ 若 }}\ x\not \in \mathbb {Z} .\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzNnEoNC1agaQo2ePatmPagiOatC3zDBCzqs2a2e2ajKQyji0zjw3)
至於小數部分
,自變量取相反數會使小數部分變成關於1的「補數」:
![{\displaystyle \{x\}+\{-x\}={\begin{cases}0,&{\text{ 若 }}\ x\in \mathbb {Z} ,\\1,&{\text{ 若 }}\ x\not \in \mathbb {Z} .\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Azta0zgzFnthBzAdAnti3aAe4nqnAzNK3oAiNoqeQaNFDzjw5nDs5)
上取整、下取整、小數部分皆為冪等函數,即函數疊代兩次的結果等於自身:
![{\displaystyle {\begin{aligned}{\Big \lfloor }\lfloor x\rfloor {\Big \rfloor }&=\lfloor x\rfloor ,\\{\Big \lceil }\lceil x\rceil {\Big \rceil }&=\lceil x\rceil ,\\{\Big \{}\{x\}{\Big \}}&=\{x\}.\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85ntaNntdEajm3ytlBz2zAnjoQyji4zAw4zqaOaAhFaqePa2w1oDnD)
而多個上取整與下取整依次疊代的效果,相當於最內層一個:
![{\displaystyle {\begin{aligned}{\Big \lfloor }\lceil x\rceil {\Big \rfloor }&=\lceil x\rceil ,\\{\Big \lceil }\lfloor x\rfloor {\Big \rceil }&=\lfloor x\rfloor ,\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NnghBzNm2zNm4yjw1njlFzDsPyti1oto3nAo2nqi2njCQo2aQzNlC)
因為外層取整函數實際衹作用在整數上,不帶來變化。
若
和
為正整數,且
,則
![{\displaystyle 0\leq \left\{{\frac {m}{n}}\right\}\leq 1-{\frac {1}{|n|}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EnqaPyghCzgo2nAdFotJBytBEzNlFzghAoqzEyqrEnjs5zDiPzDCP)
若
為正整數,則
![{\displaystyle \left\lfloor {\frac {x+m}{n}}\right\rfloor =\left\lfloor {\frac {\lfloor x\rfloor +m}{n}}\right\rfloor ,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zDGOzjK0yqw3ajCPzjG2ntG1zjKPzqoOajK3nDa2o2wNzNrAzDhA)
![{\displaystyle \left\lceil {\frac {x+m}{n}}\right\rceil =\left\lceil {\frac {\lceil x\rceil +m}{n}}\right\rceil .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83atJCzti1yqi2zDeQyja4atdFnjzDztC3nAwNzjrCaDs5njC3yjw2)
若
為正數,則
![{\displaystyle n=\left\lceil {\frac {n}{m}}\right\rceil +\left\lceil {\frac {n-1}{m}}\right\rceil +\dots +\left\lceil {\frac {n-m+1}{m}}\right\rceil ,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OztoPnqeNo2rEothFajvCoNFDyjhCoDiOngsOatw5yjCNotFEnDrE)
![{\displaystyle n=\left\lfloor {\frac {n}{m}}\right\rfloor +\left\lfloor {\frac {n+1}{m}}\right\rfloor +\dots +\left\lfloor {\frac {n+m-1}{m}}\right\rfloor .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dagi3zghBoDmOztw5zteOygsPoDm4nAoNyjo0oNo5a2eOyqaOa2o4)
代
,上式推出:
![{\displaystyle n=\left\lfloor {\frac {n}{2}}\right\rfloor +\left\lceil {\frac {n}{2}}\right\rceil .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80nAo2zjoQoNGOoDeNaNwNyto1yqnCnte0atzCntC3ntwNnDlAaNrE)
更一般地,對正整數
,有埃爾米特恆等式:[5]
![{\displaystyle \lceil mx\rceil =\left\lceil x\right\rceil +\left\lceil x-{\frac {1}{m}}\right\rceil +\dots +\left\lceil x-{\frac {m-1}{m}}\right\rceil ,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zDeOztrAnAs5ytJFyqhFaqrFo2w2oNm0ajw2zje5oDi5oNKPytlC)
![{\displaystyle \lfloor mx\rfloor =\left\lfloor x\right\rfloor +\left\lfloor x+{\frac {1}{m}}\right\rfloor +\dots +\left\lfloor x+{\frac {m-1}{m}}\right\rfloor .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Nngw1oNK2ytnFoNzBytoOztFAzgsPzDm5zgzDaNw3zDo5ygsPyjm1)
對於正整數
,以下兩式可將上下取整函數互相轉化:
![{\displaystyle \left\lceil {\frac {n}{m}}\right\rceil =\left\lfloor {\frac {n+m-1}{m}}\right\rfloor =\left\lfloor {\frac {n-1}{m}}\right\rfloor +1,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oDC0zAs3zAzFytK5yqsNoAoOaDa1zAoOaqhBzjw2nqe1nqe0atwQ)
![{\displaystyle \left\lfloor {\frac {n}{m}}\right\rfloor =\left\lceil {\frac {n-m+1}{m}}\right\rceil =\left\lceil {\frac {n+1}{m}}\right\rceil -1.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83otaOotJFoqe1ajnDntrEytmPnqw2ntrEataOajKPati5otiNyqhD)
對任意正整數
和
,有:
![{\displaystyle \sum _{k=1}^{n-1}\left\lfloor {\frac {km}{n}}\right\rfloor ={\frac {(m-1)(n-1)+\gcd(m,n)-1}{2}},}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OatKQaAw4zNhDnDhCntmOzAvCotCPnDdEzNlEa2sNaNmNzDGPotFF)
作為特例,當
和
互質時,上式簡化為
![{\displaystyle \sum _{k=1}^{n-1}\left\lfloor {\frac {km}{n}}\right\rfloor ={\frac {1}{2}}(m-1)(n-1).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80nDmNyto2yjFEoAw5aAzDzAePyja1atrEaDnDyjK3zNnFz2oQnDw4)
此等式可以幾何方式證明。又由於右式關於
、
對稱,可得
![{\displaystyle \left\lfloor {\frac {m}{n}}\right\rfloor +\left\lfloor {\frac {2m}{n}}\right\rfloor +\dots +\left\lfloor {\frac {(n-1)m}{n}}\right\rfloor =\left\lfloor {\frac {n}{m}}\right\rfloor +\left\lfloor {\frac {2n}{m}}\right\rfloor +\dots +\left\lfloor {\frac {(m-1)n}{m}}\right\rfloor .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dyjm0zDzDa2w4zDm5nDzFzthAaAi4oNzAa2zCnjaOatFBatJEoAa3)
更一般地,對正整數
,有
![{\displaystyle {\begin{aligned}&\left\lfloor {\frac {x}{n}}\right\rfloor +\left\lfloor {\frac {m+x}{n}}\right\rfloor +\left\lfloor {\frac {2m+x}{n}}\right\rfloor +\dots +\left\lfloor {\frac {(n-1)m+x}{n}}\right\rfloor \\=&\left\lfloor {\frac {x}{m}}\right\rfloor +\left\lfloor {\frac {n+x}{m}}\right\rfloor +\left\lfloor {\frac {2n+x}{m}}\right\rfloor +\cdots +\left\lfloor {\frac {(m-1)n+x}{m}}\right\rfloor .\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qzqe3nAnBzNG2oDm3njJDzDBCzNG2zNKPzji5nDCPzge5zNo0a2rF)
上式算是一種「互反律」(reciprocity law),與§ 二次互反律有關。
高斯給出二次互反律的第三個證明,經艾森斯坦修改後,有以下兩個主要步驟。
設
、
為互異奇質數,又設
![{\displaystyle n={\frac {q-1}{2}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaqeNnjBFnqdBzDaNoNmOaNeNa2a4nqs1aAeOzja3yqe4nqvFnghA)
首先,利用高斯引理,證明勒让德符号可表示為和式:
![{\displaystyle \left({\frac {q}{p}}\right)=(-1)^{\left\lfloor {\frac {q}{p}}\right\rfloor +\left\lfloor {\frac {2q}{p}}\right\rfloor +\dots +\left\lfloor {\frac {mq}{p}}\right\rfloor },}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BnjK5aNm1nDi2zjm1nAnFzDiPzqiOngaNajiQoNG3aDrCaNo5nAsO)
同樣
![{\displaystyle \left({\frac {p}{q}}\right)=(-1)^{\left\lfloor {\frac {p}{q}}\right\rfloor +\left\lfloor {\frac {2p}{q}}\right\rfloor +\dots +\left\lfloor {\frac {np}{q}}\right\rfloor }.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zAs5njG5atsPzqdAztwPaAs2oqi1aAhFatzDoNm4yjoNytm4oqdA)
其後,採用幾何論證,證明
![{\displaystyle \left\lfloor {\frac {q}{p}}\right\rfloor +\left\lfloor {\frac {2q}{p}}\right\rfloor +\dots +\left\lfloor {\frac {mq}{p}}\right\rfloor +\left\lfloor {\frac {p}{q}}\right\rfloor +\left\lfloor {\frac {2p}{q}}\right\rfloor +\dots +\left\lfloor {\frac {np}{q}}\right\rfloor =mn.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Bnjo3zge5ngnFytzDoNe0nqw5zte5aDi4yqi4njlEzjCNaAsNzNK2)
總結上述兩步,得
![{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{mn}=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PajKOaNlBngvFoDCNyjlAaDJEnjhAoNdEzDG3aqoPoNaQagrAoAoO)
此即二次互反律。一些小整數模奇質數
的二次特徵標,可以高斯符號表示,如:
![{\displaystyle \left({\frac {2}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{4}}\right\rfloor },}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QatC0ngw0aNiPnti3ytK0nqzCatnFzNi0njFAo2a5yts1otrDnjKO)
![{\displaystyle \left({\frac {3}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{6}}\right\rfloor }.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83ytdBytK2nAaOytC5zjBCaqi4ajC0oDzBajdFotm1ztdFz2rAa2w3)
下取整函數出現於若干刻畫質數的公式之中。舉例,因為
在
整除
時等於
,否則為
,所以正整數
為質數当且仅当[11]
![{\displaystyle \sum _{m=1}^{\infty }\left(\left\lfloor {\frac {n}{m}}\right\rfloor -\left\lfloor {\frac {n-1}{m}}\right\rfloor \right)=2.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AytG4aqwQnDeOzjo0aDK1aNC4zDCNzNe0nAe3ajoNoDi4ygs1zNs1)
除表示質數的條件外,還可以寫出公式使其取值為質數。例如,記第
個質數為
,任選一個整數
,然後定義實數
為
![{\displaystyle \alpha =\sum _{m=1}^{\infty }p_{m}r^{-m^{2}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EnDmOyjK3oAi5ytnAyjG4zgs3nqhCz2w1zNC0zjmPz2oNzjC4njwN)
則衹用取整、冪、四則運算可以寫出質數公式:
![{\displaystyle p_{n}=\left\lfloor r^{n^{2}}\alpha \right\rfloor -r^{2n-1}\left\lfloor r^{(n-1)^{2}}\alpha \right\rfloor .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85agdDoNKOaDoPnta5aqzDzgsNnjiQzgaQzNw5aqvEnta5ntBCajlA)
類似還有米尔斯常数
,使
![{\displaystyle \left\lfloor \theta ^{3}\right\rfloor ,\left\lfloor \theta ^{9}\right\rfloor ,\left\lfloor \theta ^{27}\right\rfloor ,\dots }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoAzDaqi0ytG4yga4njoQats3ngvBzgvAnta5agw4oqo3zts4otm0)
皆為質數。[13]
若不疊代三次方函數,改為疊代以
為㡳的指數函數,亦有
使
![{\displaystyle \left\lfloor 2^{\omega }\right\rfloor ,\left\lfloor 2^{2^{\omega }}\right\rfloor ,\left\lfloor 2^{2^{2^{\omega }}}\right\rfloor ,\dots }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Nzqi1ajw3oNJDoNJCyjBCnqoQzgzBztK3nDvDots2otw0zNo1zjlB)
皆為質數。[13]
以質數計算函數
表示小於或等於
的質數個數。由威尔逊定理,可知
![{\displaystyle \pi (n)=\sum _{j=2}^{n}\left\lfloor {\frac {(j-1)!+1}{j}}-\left\lfloor {\frac {(j-1)!}{j}}\right\rfloor \right\rfloor .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzAhEyti1aAi2oAhFoqvAnDnFzNnBa2vCnjaQygdBntK1aNC0oNnB)
又或者,當
時:[15]
![{\displaystyle \pi (n)=\sum _{j=2}^{n}\left\lfloor {\frac {1}{\sum _{k=2}^{j}\left\lfloor \left\lfloor {\frac {j}{k}}\right\rfloor {\frac {k}{j}}\right\rfloor }}\right\rfloor .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dyje0aDC5yqi5oto2zqzFoAvCaqrBzAa4ytK5aArFnDwNotnCatzC)
本小節的公式未有任何實際用途。[16][17]
- 对于所有实数x,有:
![{\displaystyle \left\lbrack {\frac {x}{2}}\right\rbrack ={\frac {1}{4}}((-1)^{[x]}-1+2[x])}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aNlCnDo3nAaOnDa4zNi0ntKNntaOotG1agnCnAvAyjC4ntdByqrB)
![{\displaystyle \left\lbrack {\frac {x}{3}}\right\rbrack ={\frac {-2}{\sqrt {3}}}\sin({\frac {2\pi }{3}}[x]+{\frac {\pi }{3}})+1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BajsOz2aNzqePzjrDa2w4ajvFoDw5nAoQnts2ajo5nDsOztdAzgoO)
- 设x为一个实数,n为整数,则
![{\displaystyle \sum _{k=0}^{n-1}E(x+{\frac {k}{n}})=E(nx)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaqiOaNa3a2wOnqs0zNi4zjC2aNe2zDmQntC5njrBagiQygoNaDJD)
![{\displaystyle E({\frac {1}{n}}E(nx))=E(x)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dnga3zDKOyqw5oqeNytCNzNwNngiNoDCPa2zDaNJDa2vEotm0oNwQ)
- 如果x为整数,则
![{\displaystyle E(x)+E(-x)=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaAhBzDiQaNK5nDzDnjwPnts1otC0ytoPaNmOzNw5aqdEzja1zDdB)
- 否则
![{\displaystyle E(x)+E(-x)=-1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PagvAnqzEnDdDzqePoAo0aNCNzjrDnja5oDdFoNo0aAhBoAvFzjm2)
截尾函数