Lower bounds on the lengths of double-base representations
Abstract
A double-base representation of an integer n is an expression n = n_1 + ... + 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.
Cite
@article{arxiv.1001.4133,
title = {Lower bounds on the lengths of double-base representations},
author = {Vassil S. Dimitrov and Everett W. Howe},
journal= {arXiv preprint arXiv:1001.4133},
year = {2011}
}
Comments
8 pages, LaTeX. Added DOIs for most references; corrected a minor error in arithmetic; made small copy-editing changes. To appear in Proc. Amer. Math. Soc