A dream problem

Not infrequently I dream that I am thinking about a mathematical problem. Usually it turns out that the problem is worthless; occasionally it is interesting but tough. A few days ago I had the unusual experience of a simple but meaningful problem.

Problem: Is there a 5-digit number, with all digits different, so that its double uses the other five digits?

There is such a number, but I will leave you the pleasure of finding it.

But of course that led to further thoughts:

• Find all such numbers. This is surely a programming exercise, and I have been far too busy to do it. (Besides, the School just replaced my defunct laptop with a Windows machine, and apart from other problems I can’t get it to install GAP.)
• For which even bases other than ten do such numbers exist? Clearly not for base 2. For base 4, the number 12 does the job, and also (if you allow 0 as an initial digit) 03. At this point I wondered if all even numbers except 2 would work; but my hand calculations suggest that 6 does not work. So the question is entirely open. Unlike the earlier ones, this is a true mathematical problem.