# 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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### Infinite numbers of coprime pairs $k(6k-1)$.

This question leads to the following consideration. Are there infinite number of pairs of the form $\big(j(6j-1),\, k(6k−1)\big)$ where $j<k$ such that $j(6j-1)$ and $k(6k−1)$ are coprime? One ...
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### Equivalent integral quadratic forms properly represent the same integers

Definitions: An integral quadratic form (IQF) is some instance of $f(x,y)=ax^2+bxy+cy^2$, where $a,b,c \in \mathbb{Z}$. Let $f(x,y),g(x,y)$ denote IQFs. We say $f(x,y)$ and $g(x,y)$ are properly ...
1 vote
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### PHI function to list relative primes

I am using a website called dcode to input numbers into the PHI function, and then receive an output of numbers relatively prime with my input. The website, unfortunately, limits output to just 500 ...
41 views

### Expressing any even natural number as a sum of primorials with coefficients

I'm having a hard time trying to solve the following problem: Given any random even natural number, $x$, prove that it can or cannot be written as the product of some integer, $b$, times the primorial ...
1 vote
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1 vote
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### Find $e$ in set closed under subtraction, hence prove co-prime elements have GCD $=1$.

In the book by Andre Weil, titled : Number theory for Beginners, is given on page 4,5 theorem #II.1 regarding a set M i.e. closed under subtraction. There are also two corollaries to the same, ...
50 views

### Simple proof that there exist that you can obtain difference of 1 between multiples of coprimes $a$ and $b$

Given two coprimes $a$ and $b$ (assume wlog that $a < b$), there are non-negative integers $n_a$ and $n_b$ such that $n_b \cdot b = n_a \cdot a + 1$. Easy to prove using Bézout's identity, but is ...
132 views

### Distinct Mersenne numbers are relatively prime specific proof verification [duplicate]

As the title states, I'm supposed to prove for distinct primes $p_1,p_2 >2$ the primes dividing $2^{p_1}-1$ and $2^{p_2}-1$ are distinct. The Wikipedia page on Theorems about Mersenne Numbers ...
130 views

### Making sure $a$ and $b$ are relatively prime

I came across this interesting problem in the Olympiad Maths challenge practice problem, and it is really fascinating: Some $n$ numbers are selected randomly from the integers $1$ to $420$. $2$ ...
66 views

### Prove that the elements of a sequence are pairwise co-prime [duplicate]

Hi I need help on this question: Consider the sequence of positive integers an, for $n ≥ 1$, defined by $a_n = 6^{2^n} + 1$. (a) Prove that the elements of this sequence are pairwise co-prime, i.e. ...
1 vote
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### Do infinite coprime points $(x,y)$ exist satisfying $x+y=n$ where $x,y,n \in \mathbb{N}$?
I'm trying to prove infinite primes exist based upon this: For $n$ prime, every grid point on the line is coprime. Examples of $x+y=n$ for $n$ prime ...
This is question 6E from the 2019 Cambridge Mathematics Tripos paper 1A (which can be accessed at https://www.maths.cam.ac.uk/undergrad/pastpapers/files/2019/paperia_4_2019.pdf). "Let $n\geq 2$ ...