(An official version and a preprint are available online.)

A *double-base representation* of an integer *n* is an expression
*n = n _{1} + … + n_{r}*,
where the

Our methods allow us to find the smallest positive integers with no double-base representations of several lengths. In particular, we show that 105 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.

We use several computer programs to obtain the results of this paper. The
program SumOf5.c shows that every positive
integer less than 326552783 has a double-base representation of
length 5. The program Check5.c finds
all representations of 326552783 modulo 4441033200890842920 as a
sum of five integers of the form ±2^{a}3^{b}.
The Magma program Check5.magma
finishes the proof that 326552783 has no double-base representation of length 5.
And the Magma code in DoubleBase.magma
can be used to check various other results from the paper.