➕➖ mathy ➗✖️

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Density, Prime, and Squarefrees

Published on 06 January 2022

The idea of utilizing the density of primes, or double counting the amount of squarefree to obtain some desired conditions. I'm still new at this technique, and analytical number theory in general, so I want to list down the problems which could be solved using it. This list will be updated whenever I find new problems that fit the criteria. And disclaimer, the order here is not based on the difficulty!

Problems

(USAMO 2014) Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in{0, 1, \ldots n}$, then $$\min ( a, b ) > c^n \cdot n^{ \frac{n}{2} }.$$
(China TST 2015) Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.
(China Southeast 2020) Arrange all square-free positive integers in ascending order $a_1,a_2,a_3,\ldots,a_n,\ldots$. Prove that there are infinitely many positive integers $n$, such that $a_{n+1}-a_n=2020$.
(China MO 2023) Prove that there exist $C>0$, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$, if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$, such that $a_m$ is squarefree.
(USA TSTST 2011) Prove that there exists a real constant $c$ such that for any pair $(x,y)$ of real numbers, there exist relatively prime integers $m$ and $n$ satisfying the relation $$ \sqrt{(x-m)^2 + (y-n)^2} < c\log (x^2 + y^2 + 2). $$