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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Radicands of square roots of the 2020s, written in simplest radical form

As of the time of writing, the current decade is the 2020s. An interesting property of this decade is that there are 3 years that satisfy the property that the square-free part (https://oeis.org/...
William Hu's user avatar
8 votes
0 answers
329 views

Bounding a sum of reciprocals of square-free integers

(Cross-posted from MSE, as the question did not get any clear answer) Fix positive integers $k$ and $n$. Let $N_1,\dots,N_r$ be all the integers less than or equal to $n$ that are squarefree and have ...
Juan Moreno's user avatar
6 votes
2 answers
364 views

Square-free numbers in an interval

Is there an explicit estimate in the literature bounding from above the number of square-free numbers in a short interval $x<n\leq x y$? I can easily do this by means of the Selberg sieve, but I do ...
H A Helfgott's user avatar
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Square-free Mersenne numbers

Questions about prime numbers are notoriously hard. Let me ask one which may be easier: QUESTION:  are there infinitely many square-free Mersenne numbers $$ M(n)\ := 2^n-1 $$ where $\ n\in\mathbb N\ $ ...
Wlod AA's user avatar
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Justify that a certain set depending on a parameter is large

If there are some square free numbers, a finite amount of them $\{t_1,t_2,...t_k\}$ and we define the set $\mathcal{N}_L=\{l\in\mathbb{N}/L-l^2\notin\bigcup t_j\mathbb{Z}^2\}$, where $n\notin\bigcup ...
MathqA's user avatar
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Estimate $ \sum_{\substack{n\leq x\\ n\in Q\\}}\log n$ and $\sum_{\substack{n\leq x\\ n\in Q\\}} n$ where $Q$ is the square-free numbers

Let $Q$ be the set of squarefree numbers. I'd like to know estimates of following sums: $$ \sum_{\substack{n\leq x\\ n\in Q\\}}\log n \qquad\text{and}\qquad \sum_{\substack{n\leq x\\ n\in Q\\}} n. $$ ...
Beta's user avatar
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Distribution of $\{x/n^2\}$

Let $x$ be a large positive real number. Let $I$ be an interval -- say, $I=[1,\sqrt{\epsilon x}]$. Let $n$ range over the integers in $I$, or over the intersection of $I$ and an arithmetic progression ...
H A Helfgott's user avatar
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3 votes
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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 $$ Q_A(x) = x \prod_{p|A} \left(1-\frac{1}{p}\right) \prod_{p \nmid A} \left(1-\frac{1}{p^2}\right) + O(\sqrt{x}). $$ ...
Iguana's user avatar
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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,...
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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 [1], section A18 and the references cited in this book. Because ...
user142929's user avatar
2 votes
1 answer
164 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). ...
user142929's user avatar
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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 ...
user142929's user avatar
2 votes
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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 ...
Turbo's user avatar
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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 ...
Turbo's user avatar
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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, ...
Pablo's user avatar
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4 votes
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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 $C$ ...
Pietro Paparella's user avatar
3 votes
0 answers
187 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 ...
Richard Voepel's user avatar
2 votes
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"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). ...
OmnipotentEntity's user avatar
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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 ...
Dr. Pi's user avatar
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8 votes
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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. ...
Marco's user avatar
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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 ...
OriginalBBB's user avatar
13 votes
1 answer
866 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 ...
alpoge's user avatar
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4 votes
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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 $O(\sqrt{x})$. ...
tnnl's user avatar
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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 ...
user35352's user avatar
3 votes
2 answers
456 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 ...
Marc Brown's user avatar
36 votes
3 answers
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-...
Fred Daniel Kline's user avatar
18 votes
4 answers
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'...
Ewan Delanoy's user avatar
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8 votes
1 answer
584 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?
Ewan Delanoy's user avatar
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