# Questions tagged [coprime]

Use this tag for questions related to integers such that the only positive integer that divides them is 1.

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### Is there any similar solutions including $\pi$ like $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{5}+\cdots=\frac{\pi}{3\sqrt{3}}$?

First equation is very popular - there are only odd numbers. Other words, numbers, which are coprime with $2$. $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{7}+\cdots=\frac{\pi}{4}$$ ...
2answers
8k views

### How many numbers in a given range are coprime to $N$?

Is there a good algorithm for counting the numbers $x$ between $A$ and $B$ with $x$ and $N$ coprime? This is just like this question except for the range. The factorization of $N$ is known. I ...
14answers
6k views

### If $(a,b)=1$ then prove $(a+b, ab)=1$.

Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$. $(a,b)=1$ means $a$ and $b$ have no prime factors in common $ab$ is simply the product of ...
2answers
6k views

### Generating all coprime pairs within limits

Say I want to generate all coprime pairs ($a,b$) where no $a$ exceeds $A$ and no $b$ exceeds $B$. Is there an efficient way to do this?
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1answer
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### How many numbers $2^n-k$ are prime?

We are all familiar with the Mersenne primes $$M_n = 2^n-1$$ and we indeed know that there are some $M_n$ that are prime. However, it is still open whether there are infinitly many $M_n$ that are ...
3answers
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### Prove two numbers are coprime

I encountered some other problem and I found a beautiful proof here Write $1/1 + 1/2 + ...1/ (p-1)=a/b$ with $(a,b)=1$. Show that $p^2 \mid a$ if $p\geq 5$. (see Thomas Andrew's post) But I thought ...
2answers
102 views

### Disprove: If $\gcd(n,2^n-1)=1$, then $2^n-1$ is squarefree.

Disprove: If $\gcd(n,2^n-1)=1$, then $2^n-1$ is squarefree. Equivalently, if $2^n-1$ is not squarefree, then $\gcd(n,2^n-1)\neq 1.$ Exercise, which I do,says to show that statement above is false. I ...
4answers
110 views

### Show that for $x,y,z\in\mathbb{Z}$, if $x$ and $y$ are coprime, then $\exists n\in\mathbb{Z}$ such that $z$ and $y+xn$ are coprime.

Show that for $x,y,z\in\mathbb{Z}$, if $x$ and $y$ are coprime and $z$ is nonzero, then $\exists n\in\mathbb{Z}$ such that $z$ and $y+xn$ are coprime. Not sure where to start on this one. I ...
1answer
52 views

### What is the probability that 2 integers have a greatest common factor of 2?

If we pick any two positive integers at random, what is the probability that their greatest common factor is 2? I have been wondering about this problem for a while and done some work on it. I started ...
1answer
96 views

### Probability of k random integers being coprimes

In this section of the Wikipedia article on coprime integers, it is stated that: More generally, the probability of $k$ randomly chosen integers being coprime is $1/\zeta(k)$. where $\zeta$ is the ...
1answer
94 views

### Stapled sequences- set of consecutive positive integers such that no one of them is relatively prime with all of the others

A stapled sequence is defined as a set of consecutive positive integers such that no one of them is relatively prime with all of the others. When I first came across this definition, I thought it ...
2answers
325 views

### $\frac{7x+1}2, \frac{7x+2}3, \frac{7x+3}4, \ldots ,\frac{7x+2016}{2017}$ are reduced fractions for integers $x\in(0,301)$. [closed]

BdMO 2017 junior catagory Question 7. $$\dfrac{7x+1}2, \dfrac{7x+2}3, \dfrac{7x+3}4, \ldots ,\dfrac{7x+2016}{2017}$$ Here $x$ is a positive integer and $x < 301$. For some values of $x$ it is ...
2answers
98 views

### Count Integers Not Greater Than $a$ Coprime To $b$

I'd like to ask how to count $f(a,b)$, the number of integers not greater than $a$ which are coprime to a given number $b$. Can $f$ be expressed using Euler's totient function?
2answers
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1answer
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### Relatively prime numbers are prime

The problem is to find all numbers $n$ such that all numbers $k>1$ smaller than $n$ and coprime with $n$ are prime.
2answers
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### Find $X, Y \in \mathbb{Z}$ such that $2^a X + (2^b - 1) Y = 1$ (coprimality)

I've been wrecking my brain trying to solve this exercise. Is this answer wrong? $$X= (2^{a})^{b-1}, Y= (-1) (2^b +1) \ [(2^b -1)(2^b +1)]^{a-1}$$
1answer
46 views

### Number of integers coprime to a given integer $q$ in some range $[x, x+y]$

I am asked to show that for $1 \leq x,y$ and an integer $q$, we have: $S(x,x+y,q) = |\{x < n \leq x + y \mid n \text{ is comprime to } q\}| = \frac{\phi(q)y}{q} + O(2^{\omega(q)})$, where: \$\...