[matthewh (OP)]

Jack sent us this:

 With a half-life of 17 years, the radioactive isotope of Promethium should have all decayed away by now.  So why does some still exist?

Let's not let the question decay away with the isotope!

[evan_au]

Uranium 235 can decay by nuclear fission after absorbing a neutron - it breaks into two nuclei (plus a couple of neutrons) that "add up" to the components of the original uranium nucleus.

There are a wide range of radioactive and stable isotopes produced by fission. One of these is promethium. Overall, the natural rate of production equals the natural rate of decay, with a worldwide natural stock estimated at around half a kilo.

Commercially, Promethium is produced in nuclear reactors, which bombard enriched Uranium 235 with neutrons.
It can also be produced by bombarding neodymium with neutrons in a nuclear reactor.
See: https://en.wikipedia.org/wiki/Promethium

[Janus]

Jack sent us this:

 With a half-life of 17 years, the radioactive isotope of Promethium should have all decayed away by now.  So why does some still exist?

Let's not let the question decay away with the isotope!

There are two answers to this question.   The first is that short-lived isotopes can be in the decay chain of long-lived isotopes.
For example, Uranium 238 has a half-life of 4.5 billion years, but decays into Thorium 90 with a half-life of 24.1 days, which in turn decays into Protoactinium 234 with a half-life of 1.17 min.  This then decays into U 234 with a half-life of 250,000 y.

Since you are always going to have U 238 atoms decaying from time to time, you will always have atoms of the two short-lived isotopes mixed in.

The other reason is that a Half-life is a statistical value.   While you have a large sample of atoms to work with, it tells you how long it will take for 1/2 of those atoms to decay, and then how long it would take for half of the remaining atoms to decay...
But it can never tell you how long it will take for a single atom to decay, only the odds that it will have decayed over a given period.  Thus any given atom can decay in the next moment, or 100 billion years from now, even if it half-life is 17 y.

So you can never really say when a given amount of an isotope will have all decayed. (though you can determine when the number of expected remaining atoms becomes insignificant.)