  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.
