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

2019/01/30 by de la Bretèche, Régis, Kurlberg, Pär, Shparlinski, Igor E.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1901.10694

Abstract

We study some counting questions concerning products of positive integers u₁, …, 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 ℚ and improve and generalize a result of N. Rome (2017).

Related

• Power-saving error terms for the number of D₄-quartic extensions over a number field ordered by discriminant
• Squares in Polynomial Product Sequences
• Bounds on the number of Diophantine quintuples
• Counting multi-quadratic number fields of bounded discriminant
• Products of shifted primes simultaneously taking perfect power values