# On the number of products which form perfect powers and discriminants of multiquadratic extensions

## TL;DR

This paper investigates counting problems related to products of positive integers forming perfect powers and extends previous results by providing asymptotic formulas for such counts and for discriminants of multiquadratic number fields.

## Contribution

It offers new asymptotic formulas for counting products forming perfect powers and for discriminants of multiquadratic extensions, generalizing earlier work by Tolev and Rome.

## Key findings

- Derived an asymptotic formula for counting perfect power products.
- Improved bounds and generalizations for discriminants of multiquadratic fields.
- Extended previous results to broader classes of number-theoretic objects.

## Abstract

We study some counting questions concerning products of positive integers $u_1, \ldots, u_n$ which form a non-zero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields which are multiquadratic extensions of $\mathbb{Q}$ and improve and generalize a result of N. Rome (2017).