Products of integers with few nonzero digits

2021/12/06 by Kaneko, Hajime, Stoll, Thomas
#11A63 #11B83 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2112.03077

Abstract

Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k,ℓ,m, the Diophantine system s(ab)=k, s(a)=ℓ, and s(b)=m, in odd integer variables a,b. When k=2 or k=3, we establish a bound on ab in terms of ℓ and m. While such a bound does not exist in the case of k=4, we give an upper bound for min\a,b\ in terms of ℓ and m.

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