Products of Two Integers Avoiding Perfect Powers

Abstract

For integers $d\geq 3$, let $F_{2,d}(n)$ be the largest size of a subset of $[n]$ containing no two distinct elements whose product is a perfect $d$-th power, and let $f_{2,d}(n)$ denote the analogous quantity when the two elements need not be distinct. Fleiner, Juh\'asz, K\"ov\'er, Pach, and S\'andor proved that both complements have order $n^{2/3}$ when $d=3$, and asked for a leading constant. They also asked whether, more generally, $n-F_{k,d}(n)$ and $n-f_{k,d}(n)$ have order $n^{k/d}$ for $1<k<d$. We establish asymptotic formula in the case $k=2$ for every fixed $d\geq3$, \[ n-F_{2,d}(n)\sim n-f_{2,d}(n) \sim C_d\, n^{2/d}(\log n)^{d-3}, \] where $C_d>0$ is given explicitly by an Euler product and a polytope volume. In particular, the extra logarithmic factor gives a negative answer to the second question for every $d\geq4$. For $d=3$ we obtain \[ C_3=\frac{\pi^2}{4} \prod_p\left(1-\frac3{p^2}+\frac2{p^3}\right), \] which answers the first question. The proof uses an exact decomposition into complementary $d$-free kernel classes, a squarefree sieve in multiplicative boxes, and a two-height polytope calculation.