Questions tagged [ceiling-and-floor-functions]

This tag is for questions involving the greatest integer function (or the floor function) and the least integer function (or the ceiling function).

The greatest integer function, or the floor function, is usually denoted by $\lfloor\_\rfloor$ (although some authors prefer $[\_]$). For a real number $x$, $\lfloor x\rfloor$ is the largest integer that is less than or equal to $x$. For example, $\lfloor 2^{1000}\rfloor=2^{1000}$, $\lfloor\sqrt{105}\rfloor=10$, and $\lfloor -\pi\rfloor =-4$.

The least integer function, or the ceiling function, is usually denoted by $\lceil\_\rceil$. For a real number $x$, $\lceil x\rceil$ is the smallest integer that is greater than or equal to $x$. For example, $\lceil 2^{1000}\rceil=2^{1000}$, $\lceil\sqrt{105}\rceil=11$, and $\lceil -\pi\rceil =-3$.

1954 questions
97
votes
2 answers

Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers?

Our algebra teacher usually gives us a paper of $20-30$ questions for our homework. But each week, he tells us to do all the questions which their number is on a specific form. For example, last week it was all the questions on the form of $3k+2$…
CODE
  • 4,601
  • 2
  • 21
  • 50
63
votes
3 answers

Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?

I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any positive irrational numbers $a\ne e$ such that…
59
votes
1 answer

Is there a "good" reason why $\left\lfloor \frac{n!}{11e}\right\rfloor$ is always even?

(A follow-up of sorts to this question.) The quantity $\left\lfloor \frac{n!}{11e}\right\rfloor$ is always even, which can be proved as follows. Using the sum for $\frac{1}{e}$, we split the fraction up into three parts: $A_n=\sum_{k=0}^{n-11}…
48
votes
4 answers

Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$. Challenge: Prove $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}.$$ I have…
47
votes
4 answers

$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

I found the following relational expression by using computer: For any natural number $n$, $$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$ Note that $\lfloor x\rfloor$ is the largest integer not…
mathlove
  • 122,146
  • 9
  • 107
  • 263
44
votes
3 answers

Does this pattern continue $\lfloor\sqrt{44}\rfloor=6, \lfloor\sqrt{4444}\rfloor=66,\dots$?

By observing the following I have a feeling that the pattern continues. $$\lfloor \sqrt{44} \rfloor=6$$ $$\lfloor \sqrt{4444} \rfloor=66$$ $$\lfloor \sqrt{444444} \rfloor=666$$ $$\lfloor \sqrt{44444444} \rfloor=6666$$ But I'm unable to prove it.…
39
votes
2 answers

there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$

recent conjecture :Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that :there exist infinite many $n\in\mathbb{N^{+}}$ such that $$S_n-[S_n]<\dfrac{1}{n^2}$$ where $[x]$ represents the largest integer…
34
votes
5 answers

How do the floor and ceiling functions work on negative numbers?

It's clear to me how these functions work on positive real numbers: you round up or down accordingly. But if you have to round a negative real number: to take $\,-0.8\,$ to $\,-1,\,$ then do you take the floor of $\,-0.8,\,$ or the ceiling? That is,…
Mirrana
  • 8,515
  • 33
  • 80
  • 114
31
votes
3 answers

$\lim_{n\to\infty}\frac{n -\big\lfloor\frac{n}{2}\big\rfloor+\big\lfloor\frac{n}{3}\big\rfloor-\dots}{n}$, a Brilliant problem

I encounter a question when visiting Brilliant: Find $\space\space\space\space\lim_{n\to\infty}s_n$ $=\lim_{n\to\infty}\frac{n - \big \lfloor \frac{n}{2} \big \rfloor+ \big \lfloor \frac{n}{3} \big \rfloor - \big \lfloor \frac{n}{4} \big \rfloor +…
31
votes
7 answers

Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $

I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one: DEFINITION Given $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that…
Pedro
  • 116,339
  • 16
  • 202
  • 362
26
votes
8 answers

What is the mathematical notation for rounding a given number to the nearest integer?

What is the mathematical notation for rounding a given number to the nearest integer? So like a mix between the floor and the ceiling function.
Rayreware
  • 862
  • 1
  • 5
  • 14
26
votes
6 answers

How to prove floor function inequality $\sum\limits_{k=1}^{n}\frac{\{kx\}}{\lfloor kx\rfloor }<\sum\limits_{k=1}^{n}\frac{1}{2k-1}$ for $x>1$

Let $x>1$ be a real number. Show that for any positive $n$ $$\sum_{k=1}^{n}\dfrac{\{kx\}}{\lfloor kx\rfloor }<\sum_{k=1}^{n}\dfrac{1}{2k-1}\tag{1}$$ where $\{x\}=x-\lfloor x\rfloor$ My attempt: I try use induction prove this inequality. It is…
math110
  • 1
  • 15
  • 119
  • 475
24
votes
4 answers

How to prove that $\frac{(5m)!(5n)!}{(m!)(n!)(3m+n)!(3n+m)!}$ is a natural number?

How to prove that $$\frac{(5m)! \cdot (5n)!}{m! \cdot n! \cdot (3m+n)! \cdot (3n+m)!}$$ is a natural number $\forall m,n\in\mathbb N$ , $m\geqslant 1$ and $n\geqslant 1$? If $p$ is a prime, then the number of times $p$ divides $N!$ is…
24
votes
3 answers

How prove this $\{a\}\cdot\{b\}\cdot\{c\}=0$ if $\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor nc\rfloor$

Interesting problem Let $a,b,c$ be real numbers such that $$\lfloor na\rfloor+\lfloor nb\rfloor=\lfloor nc\rfloor$$ for all postive integers $n$. Show that: $$\{a\}\cdot\{b\}\cdot\{c\}=0$$ where $\{x\}=x-\lfloor x\rfloor$ My partial…
math110
  • 1
  • 15
  • 119
  • 475
22
votes
0 answers

$\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square

Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$? From this question we know the condition under which $\lfloor a\rfloor\lfloor…
1
2 3
99 100