Lower bounds on the lengths of double-base representations
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- by Vassil S. Dimitrov and Everett W. Howe PDF
- Proc. Amer. Math. Soc. 139 (2011), 3423-3430
Abstract:
A double-base representation of an integer $n$ is an expression $n = n_1 + \cdots + n_r$, where the $n_i$ are (positive or negative) integers that are divisible by no primes other than $2$ or $3$; the length of the representation is the number $r$ of terms. It is known that there is a constant $a >0$ such that every integer $n$ has a double-base representation of length at most $a\log n / \log \log n$. We show that there is a constant $c>0$ such that there are infinitely many integers $n$ whose shortest double-base representations have length greater than $c\log n / (\log \log n \log \log \log n)$.
Our methods allow us to find the smallest positive integers with no double-base representations of several lengths. In particular, we show that $103$ is the smallest positive integer with no double-base representation of length $2$, that $4985$ is the smallest positive integer with no double-base representation of length $3$, that $641687$ is the smallest positive integer with no double-base representation of length $4$, and that $326552783$ is the smallest positive integer with no double-base representation of length $5$.
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Additional Information
- Vassil S. Dimitrov
- Affiliation: Center for Information Security and Cryptography, University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada
- Email: dimitrov@atips.ca
- Everett W. Howe
- Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92121-1967
- MR Author ID: 236352
- ORCID: 0000-0003-4850-8391
- Email: however@alumni.caltech.edu
- Received by editor(s): August 27, 2010
- Published electronically: February 9, 2011
- Communicated by: Ken Ono
- © Copyright 2011 American Mathematical Society and the Institute for Defense Analyses
- Journal: Proc. Amer. Math. Soc. 139 (2011), 3423-3430
- MSC (2010): Primary 11A67; Secondary 11A63
- DOI: https://doi.org/10.1090/S0002-9939-2011-10764-0
- MathSciNet review: 2813374