# 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.

14 questions
148 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, ...
50 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 $C$ ...
131 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 ...
99 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). ...
162 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 ...
638 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. ...
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 ...
655 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 ...
461 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 $O(\sqrt{x})$. ...
146 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 ...
439 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 ...
4k 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-...
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'...
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?