On the number of non-zero digits of integers in multi-base representations

Bertók, Csanád; Hajdu, Lajos; Luca, Florian; Sharma, Divyum

doi:10.5486/pmd.2017.7562

Publ. Math. Debrecen

2017

DOI: 10.5486/pmd.2017.7562

|View full text |Cite

On the number of non-zero digits of integers in multi-base representations

Csanád Bertók¹,

Lajos Hajdu²,

Florian Luca³

et al.

Abstract: Abstract. We prove various finiteness theorems for integers having only few non-zero digits in different multi-base representations simultaneously.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2020

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

(1 citation statement)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Questions such as: does every integer have a multi-base representation, or: what is the smallest number that cannot be represented in a certain system, are also of great interest; cf. [4,5,6,19]. The number of multi-base representations has also been analyzed; see [17,18].…”

mentioning

confidence: 99%

On the minimal Hamming weight of a multi-base representation

Krenn

Suppakitpaisarn

Wagner

2020

Journal of Number Theory

View full text Add to dashboard Cite

Given a finite set of bases b1, b2, . . . , br (integers greater than 1), a multi-base representation of an integer n is a sum with summands db α 1 1 b α 2 2 · · · b αr r , where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer n, i.e., the minimal number of nonzero summands in a representation of n, is log n/(log log n). This is independent of the number of bases, the bases themselves, and the digit set.For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases; for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity; thereby improving the existing bounds and closing the gap in the order of magnitude. This implies also that the greedy algorithm terminates after O(log n/ log log n) steps, and that this bound is sharp.

show abstract

mentioning

confidence: 99%

On the minimal Hamming weight of a multi-base representation

Krenn

Suppakitpaisarn

Wagner

2020

Journal of Number Theory

View full text Add to dashboard Cite

show abstract

Resolution of the equation $(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$

Liptai¹,

Németh²,

Soydan³

et al. 2020

Rocky Mountain J. Math.

View full text Add to dashboard Cite

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

On the number of non-zero digits of integers in multi-base representations

Abstract: Abstract. We prove various finiteness theorems for integers having only few non-zero digits in different multi-base representations simultaneously.

Cited by 2 publications

References 21 publications

On the minimal Hamming weight of a multi-base representation

On the minimal Hamming weight of a multi-base representation

Resolution of the equation $(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$

Contact Info

Product

Resources

About