Square-full values of quadratic polynomials

TL;DR

This paper investigates the distribution of square-full values of quadratic polynomials, establishing upper bounds on their counting function and discussing implications under the $abc$ conjecture.

Contribution

It proves an upper bound for the count of square-full values of admissible quadratic polynomials, improving understanding of their distribution.

Findings

Upper bound $S_f^{lacksquare}(N) \\ll N^{\varpi+\varepsilon}$ for some $\varpi<1/2$

Expected tighter bounds under the $abc$ conjecture

Extension of known results from linear to quadratic polynomials

Abstract

A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by $S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)$. We have known that for a relatively prime pair $(a,b)\in\mathbb N\times \mathbb N\cup\{0\}$ with a linear polynomial $f(x)=ax+b$, its counting function is $\asymp_{a,b} N^\frac{1}{2}$. Fix $\varepsilon>0$, for an admissible quadratic polynomial $f(x)$, we prove that $$S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)\ll_{\varepsilon, f} N^{\varpi+\varepsilon}$$ for some absolute constant $\varpi<1/2$. Under the assumption on the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon,f}(N^\varepsilon)$.