Lower bounds on the lengths of double-base representations

TL;DR

This paper investigates the minimal lengths of double-base representations of integers, establishing lower bounds for infinitely many numbers and identifying the smallest integers lacking representations of certain lengths.

Contribution

It introduces new lower bounds on the lengths of double-base representations and determines the smallest integers without representations of lengths 2 to 5.

Findings

Existence of infinitely many integers with longer double-base representations than previously known

Identification of the smallest integers with no double-base representations of lengths 2 to 5

Establishment of a lower bound proportional to log n / (log log n log log log n) for some integers

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.