The function that maps a real number $x$ to the largest integer less than or equal to $x$ (often denoted by $\lfloor x\rfloor$). See also (ceiling-function).

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13 views

How to pigeonhole the primes between $p_n$ and $p_{n+1}^2$ for twin prime conjecture?

For any full list of the primes up to the $n$th prime: $P = \{2, 3,5,\dots, p_n\}$, any natural number $q$ such that $ p_n \lt q \lt p_{n+1}^2$ that is not sieved by a prime in $P$ is also a prime. ...
2
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1answer
72 views

The floor function's relationship with odd and even functions

Question Suppose $f: \Re \to \Re$ is a real valued function defined on the whole real line. For each a) and b) determine if the statement is correct and justify your answer. a) If $f(x)$ is even ...
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2answers
58 views

Composing Even and Odd Functions With Floors

I have a question set that is asking me to determine if the following two statements are true and justify my answer. If $f(x)$ is even then $g(x)=[f(x)] $is even and If$ f(x)$ is odd then$ ...
0
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0answers
41 views

For any arithmetic progression $n \in \Bbb{N} : n \equiv b \pmod a$, the natural density is $\frac{1}{a}$?

This question comes from here (page 10). Given that $d(A) := \lim_{x\to\infty}\frac{1}{x}\sharp\{n \leq x : n \in A\}$, how do I get that: 1) $d(n \equiv b \pmod a) = \lim_{x\to\infty}(\left [ ...
3
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1answer
46 views

Solving Equations Containing Floor Functions

Recently I have been struggling with a problem involving the floor function. The problem is: $$ \lfloor x+5 \rfloor = 3\lfloor x\rfloor-1 $$ I have had a similar question to this however it only ...
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6answers
58 views

Why does $\lfloor x \rfloor \leq n \iff x < n+1$ but $\lfloor x\rfloor < n \iff x< n$?

$$\lfloor x \rfloor \leq n \iff x < n+1, \\\\\\\ \lfloor x\rfloor < n \iff x <n .$$ These are the two inequalities given by my book. But why are they so? Suppose $x = 2.3\quad \& ...
2
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1answer
38 views

Find this sum $\sum_{k=1}^{2015}k\lfloor \log_{2}{k}\rfloor\equiv \pmod {1000}$ [closed]

Find $$\sum_{k=1}^{2015}k\lfloor \log_{2}{k}\rfloor\equiv \pmod {1000}$$ consider $2^{i-1}\le k\le 2^i-1$,then $\lfloor\log_{2}{k}\rfloor=i-1$,This problem have simple methods?
2
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2answers
22 views

Then the value of $ [f(2)] $ where [.] represents the greatest integer function is?

A differentiable function f is satisfying the relation $$f(x+y) = f(x) + f(y) + 2xy(x+y) - \dfrac{1}{3} $$ $ \forall $ $ x , y $ belongs to $\Re$ and $$lim_{h \to 0} \dfrac{3f(h)-1}{6h} = ...
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1answer
61 views

Can this expression: $\left \lfloor \left(\frac x 2 \right)^2\right \rfloor $ be rewritten without the floor part?

I was working on a graph theory problem that asks the maximum amount of edges on a bipartite graph of $x$ vertices, I got to the conclusion it should be: $$\left \lfloor \left(\frac x 2 ...
4
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1answer
55 views

Maximum Value of a Floor-y Function

If $x$ is an integer, find the maximum value of $$f(x)=x-\left(\lfloor r_1x\rfloor+\lfloor r_2x\rfloor+\lfloor r_3x\rfloor+...+\lfloor r_{n-2}x\rfloor+\lfloor r_{n-1}x\rfloor+\lfloor ...
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1answer
37 views

$[\tan x]^2+\tan x-a$ [closed]

What is the number of integral values of $a$,$a\in(6,100)$ for which the equation $[\tan x]^2+\tan x-a=0 $ has real roots, where [.] denotes greatest integer function. My try:$[\tan x]^2+\tan ...
4
votes
1answer
56 views

$\lfloor x^k \rfloor \equiv m \pmod{n}$ with $x$ irrational

Let $x>1$ be an irrational number, and $n$ a positive integer. Is it true that, for each integer $m$, there exists an integer $k$ such that $$ \lfloor x^k \rfloor \equiv m \pmod{n}? $$
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0answers
32 views

Minimizing a sum given variables

I have this expression $$ax - b\left\lfloor\frac{cx}{m}\right\rfloor$$ Variables $a, b, c, m$ are known/given positive integers, and $x$ is an unknown integer with bounds $1 \leq x \leq m-1$. I ...
1
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2answers
57 views

How to solve $\lfloor \sqrt{(k + 1)\cdot2009} \rfloor = \lfloor \sqrt{k\cdot2009} \rfloor$

Is there any way to solve this equation (or to tell how many solutions are there), other than checking all 2009 possibilities? $\lfloor \sqrt{(k + 1)\cdot2009} \rfloor = \lfloor \sqrt{k\cdot2009} ...
2
votes
1answer
38 views

Closed form for a floor sum

I want to compute $$\sum_{i=a}^b \left\lfloor \frac{i}{k} \right\rfloor$$ Where $k < b < \infty$, and $a > 0$. I don't know where to begin (or if there's a closed form, for that matter), so ...
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3answers
84 views

Challenging $\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}$ for $\epsilon=\frac{1}{2}$.

Consider the (incorrect) claim that $$\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}.$$ How might I find the largest $\delta$ such that I can challenge $\epsilon = 1/2$? Clearly ...
3
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2answers
62 views

Maximum and minimum of a sum involving floor functions of rational numbers (contest question)

This question originates from the 1996 Canada National Olympiad. Let $r_1, r_2, \dots, r_m$ be a given set of $m$ positive rational numbers such that $\sum\limits^{m}_{k=1}{r_k} = 1 \tag{1}$ ...
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2answers
50 views

Left and Right limits of $\lfloor\lfloor x\rfloor\rfloor$ at $x = 0$

I'm self-studying calculus from Larson's Calculus 8E and on page 102 and I don't understand why $$\lim_{x\to 0^-} \frac{f(x) - f(0)}{x-0} = \frac{\lfloor\lfloor x\rfloor\rfloor - 0}{x} = \infty $$ ...
3
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1answer
50 views

Solving recurrence equation with floor and ceil functions

I have a recurrence equation that would be very easy to solve (without ceil and floor functions) but I can't solve them exactly including floor and ceil. \begin{align} k (1) &= 0\\ k(n) &= ...
0
votes
1answer
22 views

Possible value of $x$ so that fractions are in simplest form.

Which of the following could be the possible value of $x$ for which each of the fractions is in its simplest form, where $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to ...
3
votes
2answers
31 views

Prove the following floor function identity

The identity is this: $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ I am truly stumped. Help is appreciated. Thank you!
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3answers
125 views

What is $\left\lfloor0.\overline{9}\right\rfloor$? [duplicate]

We know that $0.\overline{9} = 1$ but then what is $\left\lfloor0.\overline{9}\right\rfloor$? My thought process went: $0.\overline{9} = 1$ so therefore $\left\lfloor0.\overline{9}\right\rfloor = ...
1
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2answers
40 views

Proving basic floor function inequality: $-1 \lt \lfloor 2x \rfloor - 2 \lfloor x \rfloor \lt 2$

As a direct consequence of the definition of $\lfloor x \rfloor $ I know that $$2x-1 \lt \lfloor 2x \rfloor \le 2x$$ and $$2x-2 \lt 2\lfloor x\rfloor \le 2x$$ How can I use these to show that $-1 \lt ...
1
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1answer
46 views

floor function problems

There are two parts of my problem. Given $n$ and $x$, $\lfloor \frac nx \rfloor = q$, what is the maximum possible value for $x$ such that we obtain the same floor value? i.e. $\lfloor \frac nx ...
4
votes
1answer
27 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
0
votes
1answer
24 views

How many coins do we need to get $k$ amount

In the far away land of coinsville, they use $4$ different coins as currency, $\{1,10,100,200\}$ What is the computational class of the amount of coins (minimal!!) we need to get $k$ amount? Well, ...
4
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1answer
114 views

how to sum the floors of ratios n/k when prime factorization of n is known

According to @harald-hanche-olsen the sum of the floors of ratios of $n/k$ is approximately: $$n(\ln n-1-\ln2)<\sum_{k=2}^{n-1}\Bigl\lfloor \frac nk\Bigr\rfloor<n\ln n.$$ If the prime ...
2
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3answers
59 views

Proof for $\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$

Problem: For positive integers $n,j,k$, prove that the following holds: $$\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$$ I simply ...
0
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1answer
46 views

Show $ \left\lfloor\frac{2n}{p} \right\rfloor - 2 \left\lfloor \frac{n}{p} \right\rfloor \in \{ 0, 1 \}$

I conjecture that $$ \left\lfloor\frac{2n}{p} \right\rfloor - 2 \left\lfloor \frac{n}{p} \right\rfloor \in \{ 0, 1 \}. $$ I know that it is always nonnegative, and equals $1$ for $n < p \le 2n$, ...
3
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3answers
144 views

why is limit $\lim_{x\rightarrow 0}⌊\frac{\sin x}{x}⌋ = 0$?

I was evaluating the limit $$f(x) = \lim_{x\rightarrow 0} \left\lfloor\frac{\sin x}{x}\right\rfloor$$ and I substituted the equivalent infinitesimal $\sin(x) \sim x$, obtaining $f(x) = 1$. But on ...
0
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1answer
57 views

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge?

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge ? Note: by the brackets I mean the floor function. I tried to substitute numbers and look at the members of ...
2
votes
1answer
58 views

Check my proof of a property of the greatest integer function?

Prove that $\forall n \in \mathbb{Z}, \lfloor x + n \rfloor = \lfloor x \rfloor + n $. Proof: Let $K = \{\ k\ |\ k\in\mathbb{Z},\ k \leq x+n\}$. Then, by definition, $$ \lfloor x + n \rfloor = j ...
3
votes
3answers
67 views

$f(x)$=$\left\lfloor { x }^{ 2 } \right\rfloor -\left\lfloor x \right\rfloor ^{ 2 }$ is discontinuous for all integer values of x except only at x=1

How to prove that $f(x)$=$\left\lfloor { x }^{ 2 } \right\rfloor -\left\lfloor x \right\rfloor ^{ 2 }$ is discontinuous for all integer values of x except only at $x=1$ ? Ya,even I used intuition at ...
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1answer
62 views

Characterizing the cosets of a cycle of a finite abelian group with a linear combination of floor functions

Prelude Cconsider the finite abelian group $\mathcal G = \prod_{i=1}^A \mathbb Z_{a_i}$ and let $\mathbf s \in \mathcal G$. Let $\mathcal H = \operatorname{grp}({\mathbf s})$ be the subgroup of ...
3
votes
1answer
52 views

Find this closed form $\sum_{k=1}^{n}\left(\lfloor a_{k}\rfloor +\lfloor a_{k}+\frac{1}{2}\rfloor \right)$

Let $\dfrac{1}{a_{k}}=\dfrac{1}{k^2}+\dfrac{1}{k^2+1}+\cdots+\dfrac{1}{(k+1)^2-1}$ I need some ideas to exploit for finding the closed form of $$\sum_{k=1}^{n}\left(\lfloor a_{k}\rfloor +\lfloor ...
7
votes
3answers
70 views

Closed form for the sum $\sum_{k=0}^n a^k \left\lfloor\frac{k}{p}\right\rfloor$

I want a closed form for the sum $$S=\sum_{k=0}^n a^k \left\lfloor\frac{k}{p}\right\rfloor$$ where: $a\ne 1<p<n;\quad p\in\mathbb Z$ I know a related identity, $$\quad\displaystyle ...
4
votes
1answer
70 views

If $\lfloor x^i\rfloor =i,i=1,2,3,\cdots,n$ find the maximum of $n$

Find the maximum $n$ for which there exist a real number $x$ such that $$\lfloor x^i\rfloor =i,\quad i=1,2,3,\ldots,n.$$ $\lfloor x\rfloor =1$,then $1<x<2$, $\lfloor x^2\rfloor =2$ then ...
0
votes
2answers
42 views

Indefinite Integral of Floor Function Integration by Substitution

From knowing the anti-derivative of floor function to be x*floor(x), is it possible to find the derivative of a function contained within a floor function? The ...
0
votes
2answers
69 views

$\mathcal{I}=\int\limits_0^0 \{x\}^{\lfloor x\rfloor}\,\mathrm dx=0\textrm{ or undefined ?}$

Consider the following integral: $$\mathcal{I}=\int\limits_0^0 \{x\}^{\lfloor x\rfloor}\,\mathrm dx$$ Now, my concern is that at $x=0$, the value of the integrand is $0^0$ which is undefined. It's ...
0
votes
4answers
54 views

Prove that $n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}$

I am trying to see why it holds that, for $n \in \mathbb{N},$ $$n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}.$$ I would appreciate help to see this.
4
votes
3answers
74 views

Solving $7[x]+23\{x\}=191$

For every real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-[x]$. The number of real solutions of $$7[x]+23\{x\}=191$$ is (a) 0 $\quad$ ...
2
votes
2answers
41 views

Concerning Roots of the cubic equation $f(x)=x^3+x^2-5x-1$ and the Greatest Integer (or Floor) function

The Question I got into a rather tight corner with this question. It says: Let $\alpha, \beta, \gamma$ be the roots of $f(x)=0$, where $f(x)=x^3+x^2-5x-1$. Then, the value of ...
2
votes
1answer
29 views

Are the following expression equivalent?

I need to check wether this expressions are equal but I haven't yet learned enough about the floor function to tell (I also $$ \frac 1 4 \left (5-\cos \left(n \frac \pi 2 \right)-2 (-1)^n \left(1+ ...
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votes
2answers
36 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
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votes
1answer
46 views

The integer part of $x+1$ is the integer part of $x$ plus $1$ [closed]

How do you solve the proof: If $x$ is a real number, then: $[x+1] = [x] + 1$. For my proof, I tried to describe the interior of the argument inside the parentheses, but I was unsuccessful. Please ...
6
votes
4answers
234 views

Floor function to the base 2

I'm not a math guy, so I'm kinda confused about this. I have a program that needs to calculate the floor base $2$ number of a float. Let say a number $4$, that base $2$ floor would be $4$. Other ...
2
votes
1answer
26 views

Multiplying a floor function to a number

Is it correct to write: $\cfrac{\left\lfloor{\cfrac{\pi y^2}{3\sqrt{3}x^2}}\right\rfloor}{n} \times\sqrt{3}x =\left\lfloor\cfrac{\pi y^2}{3xn}\right\rfloor$ ?
1
vote
1answer
36 views

Integer division through multiplication by reciprocal

Please help me to understand (prove) why the following statement is true. For any natural number $w > 0$ and divisor $b \in \left[ 1, 2^w \right)$, if we define a natural number $inv(b)$ such that ...
0
votes
4answers
86 views

Prove that $\left\lceil \frac{n}{m} \right\rceil =\left \lfloor \frac{n+m-1}{m} \right\rfloor$

On a discrete mathematics past paper, I must prove that $$\left\lceil \frac{n}{m} \right\rceil = \left\lfloor \frac{n+m-1}{m} \right\rfloor$$ for all integers $n$ and all positive integers $m$. ...
0
votes
2answers
35 views

Solving an inequality involving a floor

Increasing the integer $k$, I can make the floor of $L/k$ smaller than $r$: $$\left\lfloor \frac{L}{k} \right\rfloor \lt r$$ where $L, k, r$ are positive integers, $k\leq \lfloor \frac{L}{2} ...