Sums and differences of power-free numbers

2015/05/26 by Julia Brandes, Brandes, Julia
Mathematics · #11N25 #11N35 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N25 #msc:11N35

paper · pdf · doi:10.48550/arxiv.1505.07031

arxiv created 2015/05/26 · arxiv updated 2015/05/27

Abstract

We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions a, b ∈ \mathbb N to the equations a+b=n and a-b=n, where a is k-free and b is l-free. This is the first time that this problem has been studied with distinct powers k and l.

Related

• Twins of s-free numbers
• Pairs of k-free Numbers, consecutive square-full Numbers
• On powers that are sums of consecutive like powers
• On the number of pairs of positive integers x, y ≤ H such that x² + y² + 1 is squarefree
• Pairs of square-free values of the type n²+1, n²+2