Questions tagged [fractional-part]

For questions related to the fractional part of a number.

If $r$ is a real number, we can define the floor function $$\lfloor r \rfloor = \max \{n \in \mathbb{Z} : n \le r\}$$

to be the greatest integer which is not larger than $r$. The fractional part of $r$, frequently written $\{r\}$ or $r \bmod 1$, is then defined to be $$\{r\} = r - \lfloor r \rfloor$$

The fractional part of any number is thus a non-negative real number which is strictly less than $1$. The fractional part of a number is rational if and only if the number itself is rational.

Fractional parts satisfy the inequality $$\{x+y\}\leq\{x\}+\{y\},$$ with equality iff the RHS is less than $1$. Another useful fact is that, for $x\in\mathbb R\setminus\mathbb Q$, the sequence $$a_n=\{nx\}$$ is equidistributed (and in particular, dense) in $[0,1]$.

Questions involving fractional parts can often be tagged with as well.

Reference: Fractional part.

279 questions
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Minimum values of the sequence $\{n\sqrt{2}\}$

I have been studying the sequence $$\{n\sqrt{2}\}$$ where $\{x\}:= x-\lfloor x\rfloor$ is the "fractional part" function. I am particularly interested in the values of $n$ for which $\{n\sqrt{2}\}$ has an extremely small value - that is, when…
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A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when $m,n$ are relatively prime. How can we prove…
Eric Naslund
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Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of $x$. Do you have any proof?
Olivier Oloa
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Evaluate $\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy$

I want compute this integral $$\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy, $$ where $ \left\{ x \right\} $ is the fractional part function. Following PROBLEMA 171, Prueba de a), last paragraph of page 109 and firts two paragraphs of…
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A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$

I've been asked to elaborate on the following evaluation: $$ \begin{align}\\ \displaystyle {\large\int_0^{1}} \!\cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi (\left\{1/x\right\}+1)}}}} \:\mathrm{d}x & =…
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For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$

Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. Can anyone give some hint to solve this problem? I tried contradiction but could not reach a proof. I spend part…
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Is $1$ a limit point of the fractional part of $1.5^n$?

It is an open problem whether the fractional part of $\left(\dfrac32\right)^n$ is dense in $[0...1]$. The problem is: is $1$ a limit point of the above sequence? An equivalent formulation is: $\forall \epsilon > 0: \exists n \in \Bbb N: 1 -…
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Sum of $\{n\sqrt{2}\}$

I would like to prove (rigorously, not intuitively) that $$\sum_{n=1}^N \{n\sqrt{2}\}=\frac{N}{2}+\mathcal{O}(\sqrt{N})$$ where $\{\}$ is the "fractional part" function. I understand intuitively why this is true, and that's how I came up with this…
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Equality of sums with fractional parts of the form $\sum_{k=1}^{n}k\{\frac{mk}{n}\}$

I recently encountered the following equality ($\{\}$ denotes fractional part): $$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$ and found it very interesting as most of the individual summands on one…
ruadath
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The fractional part of $n\log(n)$

When I was thinking about my other question on the sequence $$p(n)=\min_a\left\{a+b,\ \left\lfloor\frac {2^a}{3^b}\right\rfloor=n\right\}$$ I found an interesting link with the sequence $$q(n)=\{n\log(n)\}=n\log(n)-[n\log(n)]$$ the fractional part…
E. Joseph
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Solve $\frac{1}{\left\lfloor{x}\right\rfloor}+\frac{1}{\left\lfloor{2x}\right\rfloor}=\{x\}+\frac13$

Problem Statement:- Solve: $$\dfrac{1}{\left\lfloor{x}\right\rfloor}+\dfrac{1}{\left\lfloor{2x}\right\rfloor}=\{x\}+\dfrac{1}{3}$$ where $\left\lfloor{x}\right\rfloor$ denotes the integral part of $x$ and…
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$\{x^2\} = \{x\}^2$, how many solutions in interval $[1, 10]$

Find how many solutions there are in the interval $[1, 10]$ to the fractional part equation: $$\left\{x\right\}^2 = \left\{x^2\right\}$$ Where $\{\cdot\}$ is the fractional part function, meaning that: $$\left\{a\right\} = a - \left\lfloor a…
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Solving $f(x+1)=xf(x)f\left(\frac1x\right)$

Recently, while scribbling around, I came up with the functional equation $f(x+1)=xf(x)f\left(\frac1x\right)$. It is somewhat similar to the functional equation for which the Gamma function is a solution, since $\Gamma(x+1)=x\Gamma(x)$, but it has…
KKZiomek
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Limit of alternating sum of fractional parts

Find $\newcommand{\pars}[1]{\left\{ \frac{n}{#1} \right\}}$ $$\lim_{n\to\infty}\dfrac{1}{n} \left( \pars{1} - \pars{2} + ... + (-1)^{n+1} \pars{n} \right),$$ where $\left\{ x \right\} $ denotes the fractional part of $x$. My guess is that the limit…
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Does $\sum_{k=1}^n|\cot \sqrt2\pi k|$ tends to $An\ln n$ as $n\to\infty$?

Question: How can we prove that $$L(n)=\sum_{k=1}^n\left|\cot \sqrt2\pi k\right|=\Theta(n\log n)$$ as $n\to\infty$? Furthermore, if $\sqrt2$ is replaced with a quadratic irrational number, does it still holds? Numerical experiment. By plotting…
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