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Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

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Prove that $N - \lfloor{N/p}\rfloor = \lfloor{\frac{p-1}{p}\left({N + 1}\right)}\rfloor$ for positive $N$ and prime $p$

I am counting the number of positive integers less than or equal to some positive integer $N$ and not divisible by some prime $p$. This gets generalized for $k$ primes where I use the principle of ...
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Can this be solved to remove the floor function and simplify the answer?

I've been working through a derivation and have arrived at the following exprssion: $$E = 1 - \frac{x}y \left( \bigg\lfloor \dfrac{2x+yx-2}{2y} \bigg\rfloor \right)^{-1}$$ where $x,y \in \mathbb{R^+...
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Algebraic Closed Form for $\sum_{n=1}^{k}\left( n- 3 \lfloor \frac{n-1}{3} \rfloor\right)$

Let's look at the following sequence: $a_n=\left\{1,2,3,1,2,3,1,2,3,1,2,3,...\right\}$ I'm trying to calculate: $$\sum_{n=1}^{k} a_n$$ Attempts: I have a Closed Form for this sequence. $$...
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1answer
34 views

Understanding Binomial coefficient with floored terms

I was reading through the notation used in a paper on arxiv.org when I came across this on page 6: $[x]$ the floor of $x$ $\{x\}$ the sawtooth function of $x$. That is $\{x\} = x - [x]$ $\begin{...
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Is my approach correct to this equation?

The problem is the following: Does $a \in \mathbb{R}$ exist such that $[a + \sqrt{2n + 1}] = [a + \sqrt{2n + 2}]$ for all $n \in \mathbb{N}$? ($[x]$ denotes the whole part of $x$). Note: I will ...
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1answer
71 views

Double sum over $\left\lfloor{ac+bd\over k}\right\rfloor$

We have $$\left\lfloor{ac+bd\over k}\right\rfloor-\left\lfloor{ac+bd-1\over k}\right\rfloor=1-\left\lceil{ (ac+bd)\mod{k}\over k}\right\rceil$$ for $a,b,c,d,k$ - integers, $a\geqslant0$, $b\geqslant0$,...
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1answer
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Injectivity of $f(x) = x + [x/2]$, and finding an explicit inverse

Context: This question comes up as a tangent to an earlier MSE question from today. The OP of this question was, in effect, seeking an explicit inverse to the function $$f(x) = x + \left[ \frac{x}{2}...
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Identity involving floor, ceiling and nearest integer functions

For $n\geqslant0$, $m>0$, $s>t\geqslant0$, $n,m,s,t$ - integers we have $$\sum\limits_{k=0}^{m-1}\left\lfloor{n+ks+t\over ms}\right\rfloor=\left\lfloor{n+t\over s}\right\rfloor$$ $$\sum\limits_{...
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Prove that if $nm ≤ nx < nm + n $, where $ n,m ∈ ℤ $ and $ x ∈ ℝ$, then there exists …

Prove that if $nm ≤ nx < nm + n $, where $ n,m ∈ ℤ $ and $ x ∈ ℝ$, then there exists $j$ such that $j ∈ ℤ$ and $0≤ j <n$ for which $ nm+j≤ nx <nm+j+1 $. I'm trying to prove Hermite's ...
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Compute $\lim_{x\to\infty} x \lfloor \frac{1}{x} \rfloor$

I'm working out a limit and I'm not sure if my assumption is considered rigorous $$\lim_{x\to\infty} x\left\lfloor\frac1x\right\rfloor$$ I supposed that $0\leq x\left\lfloor\frac1x\right\rfloor \leq \...
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1answer
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Floor equation $\lfloor 3x-x^2 \rfloor = \lfloor x^2 + 1/2 \rfloor$

Solve the equation: $$\left \lfloor 3x-x^2 \right \rfloor = \left \lfloor x^2 + 1/2 \right \rfloor$$ In the solution it writes We notice that $x^{2}+\frac{1}{2}> 0$ therfore $\left \...
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5answers
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Apostol's Calulus: Prove that $[x+y] = [x]+[y]$ or $[x]+[y]+1$, where $[·]$ is the floor function.

Prove that $[x+y] = [x]+[y]$ or $[x]+[y]+1$, where $[·]$ is the floor function I'm Having a little bit of trouble with the last part of this proof. First, I will use the definition of floor ...
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Calculating $\lim_{x \to 0}{\frac{ \lfloor x \rfloor}{\lfloor x \rfloor}}$

My question is about $$\lim_{x \to 0}{\frac{ \lfloor x \rfloor}{\lfloor x \rfloor}}$$ where the notation is the floor function. I've graphed it and it is 1 everywhere except for $[0, 1)$. So, I ...
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1answer
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Find the global min of $\lfloor{(1/2)(\lfloor{N/p}\rfloor+3-\sqrt{(\lfloor{N/p}\rfloor+1)^2-4N})}\rfloor$

Denote this function as ${a}_{l}$. Here $p$ is prime but not necessary for the solution, just $p \ge 2$ is needed. This solution is for fixed $p$ with $N$ allowed to vary. Now a plot of this ...
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1answer
23 views

Proving that $\lfloor{i/2^h}\rfloor = \lfloor \lfloor\cdots \lfloor i/2\rfloor/2\cdots\rfloor/2\rfloor$

I am trying to prove that $$\left\lfloor{\frac{i}{2^h}}\right\rfloor$$ equals to performing a series of $h$ operations of $$\left\lfloor\frac{\left\lfloor\frac{\left\lfloor\frac{i}{2}\right\rfloor}{2}\...
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Prove that $\lfloor \sqrt{(p-1)p} \rfloor = p - 1$ and likewise $\lceil \sqrt{(p-1)p} \rceil = p$.

Here $p$ is prime but is not necessary for the problem just that $p \ge 0$. I suspect that a statement like $p-1 \le \sqrt{(p-1)p} \le p$ would be the case but I am not certain how to establish this ...
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Show that $g(x)=[\frac1x]\sin x$ has a limit in $x=0$.

Show that $g(x)=\left[\frac1x \right]\sin x$ has a limit in $x=0$. ( $[1/x]$ as greatest integer less than or equal to $1/x$) I tried to use squeeze theory to find the limit of this function, but I ...
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1answer
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What is $\int \lfloor x^n \rfloor dx$ where $n$ is any real number?

I've been trying to figure out a general rule for integrating functions of the form $\lfloor x^n \rfloor$. I don't have any ideas other than trying some kind of clever substitution but I have no idea ...
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Closed form of $\int{\lfloor{x}\rfloor}dx$

I calculated $\int{\lfloor{x}\rfloor}dx$ and i got this result: $$\int{\lfloor{x}\rfloor}dx = \frac{x^2-x}{2}+\sum_{k=1}^{\infty}\left(\frac{\sin(k\pi x)}{k\pi}\right)^2+c$$ Do you know if this series ...
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2answers
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Find a necessary and sufficient condition

Let $(a_n)_{n=1}^\infty$ be a real sequence. Find a necessary and sufficient condition for $(a_n)$ so $(\lfloor a_n \rfloor)_{n=1}^\infty$ converges to $0$. Hi everyone. I am trying to brush up on ...
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1answer
156 views

summation of floor function involving prime number

Can you give me some hints on solving following summation. Is there any theory concerning the following summation? $p$ is a prime number > 2 $$\sum_{s=2}^{p-1}\left(\left\lfloor\frac{p}{s}\right\...
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1answer
59 views

Does $\left\lfloor\frac{x^2+x}{i}\right\rfloor - \left\lfloor\frac{x^2}{i}\right\rfloor = \left\lfloor\frac{x}{i}\right\rfloor$?

Does $\left\lfloor\dfrac{x^2+x}{i}\right\rfloor - \left\lfloor\dfrac{x^2}{i}\right\rfloor = \left\lfloor\dfrac{x}{i}\right\rfloor$? where $i \le x^2$ and $x$ are any positive integer. Intuitively, ...
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1answer
134 views

Algorithm for finding sequence verifying a floor equation

We are looking for an algorithm solving the following problem. Given a sequence $ 0 < x_1< \dots < x_n $ find a sequence $0 < y_1 < \dots < y_n$ such that $\forall j \in \{2, \dots, ...
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2answers
64 views

Limit of a sequence with floor function.

How do I compute the following limit: $\lim \limits_{n \to \infty} \frac{n + \lfloor \sqrt[3]n\rfloor^3}{n - \lfloor \sqrt{n+9}\rfloor}$ Without the floor function this would be simple, but I never ...
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0answers
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Mircea Merca's conjecteture

Mircea Merca conjectured that $$\left \lfloor{\frac{1}{n}\sum_{k=1}^n\sqrt{k}}\right \rfloor=\left \lfloor{\left(\frac{2}{3}+\frac{1}{6n}\right)\sqrt{n+1}}\right \rfloor$$ John Zacharias claimed that ...
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Divisor sum simplification

How does this : $$D(n)=\displaystyle\sum\limits_{i=1}^n i \left\lfloor\frac{n}{i}\right\rfloor$$ become $$D(n)=\displaystyle\sum\limits_{i=1}^{n/(u+1)} i \left\lfloor\frac{n}{i}\right\rfloor + \sum_{d=...
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1answer
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Calculating $\sum_{k=0}^{\lfloor \frac{p}{2} \rfloor} \binom{p}{k}$

I'm trying to find the value of: $$\sum_{k=0}^{\left \lfloor \frac{p}{2} \right \rfloor} \binom{p}{k}$$ For even and odd $p$, the indication I was given suggests writing it as $$\frac{1}{2}\left (\...
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1answer
68 views

Constructing a 2-periodic extension of the absolute value function using floor and ceiling functions

I am trying to use floor and ceiling functions to construct a 2-periodic extension of the function $f(x) = |x|, -1 \leq x \leq 1$. Through trial an error I have been able to show that: $f(x) = 1 - \...
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1answer
77 views

For $s, t \in \{-\lfloor (n-1)/2 \rfloor, \dots, -1, 0, 1, \dots, \lfloor n/2 \rfloor \}$, $n \in \mathbb{Z}_{ > 0}$, show $s-t \neq \ell n$

This is an intermediate step to a problem that I don't know how to prove. Added 4/29/2019: (Abstract) algebra is not my strongest subject, but if that is necessary to tackle this problem, I'll ...
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0answers
75 views

Calculate the number of nonnegative integer solutions of $ax+by\leq c$.

If $a$, $b$, and $c$ are known, and $x$ and $y$ are integers greater than or equal to zero, how many possible values of ($x$, $y$) exist that satisfy the equation $$ax + by \le c\,?$$ I have ...
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Non-linear congruence with two variables

Find all pairs $(m,n)$ of positive integers such that $$\left\lfloor\frac{178^m}{1117}\right\rfloor\equiv178n\pmod{1116}.$$ Obviously, the congruence has a solution iff $\left\lfloor178^m/1117\...
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Equivalence relation with the floor function

Let us consider a function $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ and we define the equivalence relation $\sim$ on $\Bbb{R}$ such that: $$x\ \sim\ y\qquad\Leftrightarrow\qquad f(x)=f(y).$$ Note: ...
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1answer
33 views

Is it possible to show that $a\cdot a^{\lfloor\frac{b}{2}\rfloor}\cdot a^{\lfloor\frac{b}{2}\rfloor} = a^b$ when $b$ is odd

I have $a$ and $b$ and $b$ is odd $a$ is an integer and $b$ is a strictly positive integer. Is there a way I can show: $a\cdot a^{\lfloor\frac{b}{2}\rfloor}\cdot a^{\lfloor\frac{b}{2}\rfloor} = a^b$ ...
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1answer
65 views

Proof for Natural number Identities

I am now trying to find proof for the following, which are significant to establishing proof for the Prime number relation that was originally stated in the question I posted here: $$\Bigl \lfloor \...
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3answers
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Proving If $k \le \left\lfloor \frac{n}{2} \right\rfloor$ then $\binom{n}{k-1} < \binom{n}{k}$

So I'm trying to do a proof for this problem: If $\displaystyle{k \le \left\lfloor \frac{n}{2} \right\rfloor}$ then $$\displaystyle{\binom{n}{k-1} < \binom{n}{k}}$$ I can do it algebraically but ...
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0answers
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Inequality involving $\text{floor}(x)$ [duplicate]

I'm trying to prove the following inequality $\forall x,y \in \mathbb R$ : $$\lfloor x \rfloor + \lfloor x + y \rfloor + \lfloor y \rfloor \le \lfloor 2x \rfloor + \lfloor 2y \rfloor$$ If one thing ...
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2answers
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Prove that $\lfloor x \rfloor +\lfloor 2x \rfloor + \dots +\lfloor 32x \rfloor =12345$ has no solution [closed]

Given that $\lfloor x \rfloor +\lfloor 2x \rfloor + \dots +\lfloor 32x \rfloor =12345$ where $\lfloor x \rfloor $ denotes the floor function, prove that there is no $x$ that satisfies the equation.
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1answer
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Find $\sum_{n=1}^\infty\frac{2^{f(n)}+2^{-f(n)}}{2^n}$, where $f(n)=\left[\sqrt n +\frac 12\right]$ denotes greatest integer function

Question: Let $f(n)=\left[\sqrt n +\dfrac 12\right]$, where $[\cdot]$ denotes greatest integer function, $\forall n\in \Bbb N$. Then, $$\sum_{n=1}^\infty\frac{2^{f(n)}+2^{-f(n)}}{2^n}={\color{red}?...
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0answers
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Properties of the floor function for $a\left \lfloor{\frac{x+b}{c}}\right \rfloor$

Is there a way to split the following floor function setup into two separate terms with the x in one term and the c in another term? Such as: $a\left \lfloor{\frac{x+b}{c}}\right \rfloor$ = $a\left \...
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1answer
37 views

Describe the sequence $2,2,1,0,0,1,2,2,…$ using the floor function

Is there a way to describe the sequence $2,2,1,0,0,1,2,2,1,0,0,1,2,2...$ by using the floor function? I can describe the series using a sinusoidal function but wanted to get it in terms of a floor ...
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1answer
45 views

Difference of 2 floors vs floor of difference

Good day, we are currently covering basic principles for algorithm optimisation and we were tasked with explaining the following problem. Assume $x,y \in \Bbb R$. How much can the following two ...
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4answers
205 views

Disprove the following statement: For all real numbers $x$ and $y$, if $x + \lfloor x \rfloor = y + \lfloor y \rfloor$ then $x = y$.

Disprove the following statement: For all real numbers $x$ and $y$, if $x + \lfloor x \rfloor = y + \lfloor y \rfloor$ then $x = y$. Aka: Prove the negation: There are real numbers $x$ and $y$, that $...
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2answers
56 views

How to represent $x^y$ = [Integer] + [Remainder]?

For example: $5^{\frac{1}{2}} = 2.23606\ldots = 2 + 0.23606\ldots$ Can we do this for $x^y$ in general? Motivation: a way of expressing the floor, $ \lfloor x^y\rfloor $, of an exponential $x^y$ ...
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2answers
691 views

Proof: For all real numbers x and y if x + floor of (x) = y + floor of (y) then x = y

I am trying to determine if this statement is true or false: I think that it is true, if i let x = 2.5 then the left side is 4.5 and if i let y be anything but 2.5 then x + floor of (x) cant not ...
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1answer
42 views

is this statement about x, y, and z true?

I would like to know if the following statement about x,y and z is true: $$x=\lfloor\frac{y}{z}\rfloor \iff z=\lfloor\frac{y}{x}\rfloor$$ I think it is true but am having a hard time wrapping my ...
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4answers
298 views

Prove that $\lfloor x\rfloor \geq y$ if, and only if, $x\geq\lceil y\rceil$

I have some trouble proving that if $x,y\in\mathbb{R}$ then $\lfloor x\rfloor \geq y$ if, and only if, $x\geq\lceil y\rceil$. I have tried some different approaches, the most recent being a proof by ...
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0answers
26 views

What is the significance of the greatest integer functions (step function)? [duplicate]

I'm currently in university and recently we were discussing the greatest integer function / step function and how to find the limits of them. It was a very easy concept to get the grasp of but I was ...
3
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3answers
69 views

Proving $\lfloor f(\lfloor x\rfloor)\rfloor=\lfloor f(x)\rfloor$

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous increasing function such that $$\forall x\in\mathbb{R} \;f(x)\in\mathbb{Z}\implies x\in\Bbb{Z.}\quad (1)$$ I would like to prove that $\lfloor f(\...
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1answer
63 views

Are there ways to solve miscellaneous equations such as $\sin x=\log [x]$ without drawing the graphs?

Consider the example $$\sin x=\log [x]$$ where $[\,·\,]$ represents Greatest Integer Function. It is a miscellaneous equation, and I have been told that the only way to solve it is to draw the graphs ...
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1answer
32 views

Which of the following relations are true?Here $[x]$ denotes the greatest integer less then or equal to $x$ (as in option b) symbol

Which of the following relations are true? $1)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{\frac{n(n+1)}{2}}$ $2)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{[\frac{n}{2}]}$ $3)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{n^2}$ ...