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arXiv · 1001.4133

Lower bounds on the lengths of double-base representations

Vassil S. Dimitrov·Everett W. Howe
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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.

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Vassil S. Dimitrov, Everett W. Howe. 2010-01-23. Lower bounds on the lengths of double-base representations. https://doi.org/10.1090/s0002-9939-2011-10764-0

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