# Questions tagged [square-free]

An element of an integral domain is called *square-free* if it isn’t divisible by any non-unit square. Most notable applications are square-free integers and square-free polynomials.

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1answer
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### Explicit bounds on number of squarefree numbers coprime to a certain number

We know that the number of squarefree integers $\le x$ that are coprime to $A$ is $$Q_A(x) = x \prod_{p|A} \left(1-\frac{1}{p}\right) \prod_{p \nmid A} \left(1-\frac{1}{p^2}\right) + O(\sqrt{x}).$$ ...
1answer
102 views

### Probabilistic interpretation of square free numbers and other properties

We can use the Lindberg condition to show the distribution of number of prime divisors of an integer approaches Gaussian. Is there a similar probabilistic formulation for square free numbers? That is,...
0answers
20 views

### Asymptotic size for the number of terms not exceeding $n$ in the class $r$ for a classification of the type Erdös-Selfridge for square-free integers

It is possible to define a classification similar than the Erdös-Selfridge classification of primes for different sequences. Please ee , section A18 and the references cited in this book. Because ...
1answer
139 views

### What about a formula similar than Mill's formula, but producing positive integers without repeated prime factors?

The Wikipedia's article for Prime number shows a known and curious formula for primes from its section Formula for primes, I say the Mills' theorem (please see also the Wikipedia Mills' constant). ...
0answers
81 views

### Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers

Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...
0answers
103 views

### Queries on distribution of prime divisors by magnitude?

Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$. What is the probability distribution of ...
0answers
156 views

### Magnitude and distribution of largest prime factor?

Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors. What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number? What is ...
0answers
199 views

### Gaussian square-free moat

Is there a sequence $\{z_n\}_{n=1}^\infty$ of distinct square-free Gaussian integers with $$\sup_{n \geq 1} |z_{n+1} - z_n| < \infty ?$$ For the analogous problem with Gaussian primes instead, ...
0answers
77 views

### Rank of binary matrix related to the number of positive squarefree integers less than $n$

I posted this question at the Mathematics SE, but received no response there so I am posting it here. The following fact is stated in the comments-section of sequence A013928 in the OEIS. Let $C$ ...
0answers
159 views

### Growth Rate of the Square-Free Part

In the course of considering this Diophantine equation, I convinced myself that the following question is interesting: If $n$ is large, must it be the case that the square-free part of $2^n-1$ is ...
0answers
118 views

### “Left over factors” of Fibonacci numbers squarefree?

One fact about Fibonacci numbers is they have the Mersenne-like property that $F_{mn}$ is divisible by $F_m$ and $F_n$ (but not necessarily $F_m F_n$, $F_3^2 \nmid F_9$ is a simple counterexample). ...
0answers
173 views

### Riemann's explicit formula for square-free numbers

We know that for $x$ being a half-integer $$\sum_{n\leq x}\Lambda(n)=x-\sum_{\rho}\frac{x^\rho}{\rho}+O(1).$$ Is there a similar formula for $\sum_{n\leq x}\mu(n)^2$ in the literature? The underlying ...
2answers
835 views

### Squarefree Fibonacci Numbers

Let $F_n$, $n\geq 0$, be the sequence of Fibonacci numbers, where $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\geq 1$. A number is squarefree if it is is not divisible by the square of a prime number. ...
1answer
342 views

### Can you find squares in this class?

For a problem I am working over, I would like to prove that numbers of the following type are not squares $p(l^4+6l^2m^2-3m^4)$ where $p,l,m$ are integers an $p$ prime. I have already found various ...
1answer
789 views

### Most discriminants are almost squarefree

Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$. Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...
1answer
565 views

### Estimation of $\sum_{n \leq x} \frac{k(n)}{n}$ , with $k(n)$ the squarefree kernel

I came across a poblem where they ask you to find an estimation of $\sum_{n \leq x} \frac{k(n)}{n}$, with $k(n) = \prod_{p \mid n} p$ the squarefree kernel of $n$, with an error term of $O(\sqrt{x})$. ...
0answers
149 views

### On the number N(x,y) of those integers n<x, with squarefree core k(n)<y

I'm asking something that may be trivial for those who are deeply into Analytic Number Theory, but unfortunately I'm still not into that set. The core $k(n)$ of an integer $n$ is the product of all ...
2answers
446 views

### A set with not too many integers of the form $\alpha \beta^n + r$

Consider the following (easy) lemma. Lemma. There is a subset $Q$ of the positive integers and a fixed constant $N > 0$ such that 1)$Q$ has positive asymptotic density and 2)for each ...
3answers
5k views

### $\prod_{n=1}^{\infty} n^{\mu(n)}=\frac{1}{4 \pi ^2}$

When I tested this in Mathematica, I had expected it to say it did not converge. However, I got this: $$\prod_{n=1}^\infty n^{\mu(n)}=\frac{1}{4 \pi ^2}$$ Note: this is the reciprocal of (3) zeta-...
4answers
3k views

### are there infinitely many triples of consecutive square-free integers?

The title says it all ... Obviously, any such triple must be of the form $(4a+1,4a+2,4a+3)$ where $a$ is an integer. Has this problem already been studied before ? The result would follow from Dickson'...
1answer
513 views

### Integer polynomial (of degree >1) all of whose values are square-free

Is there an integer polynomial $A \in {\mathbb Z} [ X ]$ of degree $d\geq 2$ such that for any integer $n\in {\mathbb Z}$ , $A(n)$ is a square-free integer?