# Questions tagged [fractional-part]

For questions related to the fractional part of a number.

179 questions
2answers
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### Finding the limit of this recurrence relation

Let's say we have the equation $$a_{n+1} = Frac\left(\frac{1}{\ln(1+a_n)}\right)$$ with $a_0\ne 0$ and that it is not negative. I've tried finding the limit of $a_n$ as $n \rightarrow \infty$, but I ...
2answers
36 views

### Find the total number of solution of the equation

Find number of solution of :- $$[x]^2 = x + 2\{x\}$$ where $[.]$ represents greatest integer function and $\{.\}$ represents fractional part.
1answer
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### Counting integers with a least prime factor greater than $x$ in a sequence of $x$ consecutive integers.

It is well known from Sylvester-Schur that in any sequence of $x$ consecutive integers, there is always at least one integer divisible by a prime greater than $x$. I am interested in counting the ...
1answer
19 views

### proving floor, and rounding floats in some programming languages

Here is a simple property which is very useful while programming : In most programming languages, ...
2answers
75 views

### Counting the number of integers with their least prime factor greater than $x$ between $ax$ and $ax+x$

Let: $x \ge 2, a \ge 1$ be integers. $x\#$ be the primorial for $x$ $\mu(i)$ be the möbius function. $\text{lpf}(x)$ be the least prime factor of $x$. $p_k$ be the $k$th prime which is the highest ...
1answer
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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 ...
2answers
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3answers
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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 ...
0answers
61 views

### Trigonometric Integral Involving the fractional part

Let $\{\}$ denote the fractional part function, does the following Integral admit a closed-form ? $$\int_{0}^{\pi/2}\bigg\{\frac{1}{\cos(x)}\bigg\}dx$$
2answers
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### Integral involving the fractional part function

Let $\{\}$ denote the fractional part function and $s>1$ be a real number, then does the following integral admit a closed-form ? $$\int_{0}^{1}\bigg\{\frac{1}{x^s}\bigg\}dx\,$$
1answer
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### The functional equation $f(-x+b)=f(x)$

I can solve the (periodic) functional equation $f(x+b)=f(x)$ completely ($x\in \mathbb{R}$ and $b\neq 0$). Indeed, its general solution is $f=\phi o (\; )_b$, where $(\; )_b$ is the $b$-decimal (...
1answer
73 views

### Double integral involving fractional part

I am interested in solving the following definite double integral $$\int_0^1\int_0^1\Bigl\{\frac{1}{x}\Bigr\}\Bigl\{\frac{1}{y}\Bigr\}\frac{(1-x)(1-y)}{1-xy} dx dy$$ where $\{z\}=z-\lfloor z\rfloor$...
1answer
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### Proving that $\left\{ \frac{sm}{p} \right\} < \left\{ \frac{sn}{p} \right\} < \frac{s}{p}$ if and only if $s\not\mid p-1$

I have the following question from USAMO 2006: "Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$. Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ ...
2answers
31 views

### How to get the fractional part of a product of large numbers with machine precision?

Let us have about 100 or so random (exact) floats such as: $$A_1 = 1234123.428\\ A_2 = 13713.4193\\ A_3 = 0.1332\\ A_4 = 123.13213\\ ...$$ Now I want to find an efficient way to get the fractional ...
1answer
103 views

### Definite integral on fractional part

Prove following definite integral: $$\int_0^1\Bigl\{\frac{1}{x}\Bigr\}\ln(x)\,dx = \gamma_0+\gamma_1-1$$ Found in "Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis" ...
1answer
42 views

### Functional equation $f(a)·f(b)=\Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$

I am trying to get all possible solutions of the following functional equation: $$f(a)·f(b)= \Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$$ Where {} mean fractional part function. Solutions only need to be ...
1answer
49 views

### PDF of random variable containing fractional part $x=\{1/r\}$

What happens if we combine fractional part problems and randomness? For a start here are two examples. Let $r$ be a random variable with a given probability distribution function (PDF) $f(r)$ and ...
1answer
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0answers
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3answers
131 views

### Square root of fractional part integral

Does the following integral have a closed form ? $$\int_{0}^{1}\sqrt{\bigg\{\frac{1}{x}\bigg\}}dx$$ Where $\{x\}$ denotes the fractional part of $x$.
1answer
190 views

### Double fractional part integral

Let $\{\}$ denote the fractional part, does the following integral have a closed form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{x\,y}\bigg\}^2dx\,dy$$
3answers
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2answers
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### Find the real values of $x$ that satisfy the equation $7[x]+23\{x\}=191$

For any real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ (i.e. floor function) and $\{x\}=x-[x]$ .Then, the number of real solutions of the equation $$7[x]+23\{x\}=191$$ ...