Thanks. I know what the codes mean from the wonderful internet.

Even when the codes first appeared, there were no other symptoms of
trouble; the pump was actually running during a cold engine start,
there was no sign of pump corrosion, all the hoses looked OK.

The check engine light went out by itself about a week ago and has
stayed off. I thought one needed an OBD reader to reset these warnings.
any comments?

P1421 and P1432? (not P1423)

These codes say that the secondary air system is not working correctly. The
secondary air is the Air Pump and its plumbing that punps fresh air into the
exhaust stream immediately after the exhaust leaves the head and enters the
exhaust mainfold.

1998 E36 M3
check engine light came on a few weeks ago

Autozone code reader found P1421 & P1423: BMW specific codes

A search of the net finds that these codes indicate low oxygen in both
exhaust manifolds. Nothing serious though - the oxygen is delivered by
the secondary air pump for about 100 seconds after a cold start - now I
know what that thing is that sounds like a vacuum cleaner.

The pump is actually fine and has no signs of corrosion which can be
caused by exhaust gases coming back through a defective non-return
valve on the manifold.

So I browsed around some more, and found that a relay in the main power
distribution box can be bad, giving these codes. Pulled it and
re-installed the same one, went to Autozone and had the codes reset.

Codes came back a few weeks later - pump still hums normally during a
cold start, so I know it's not a serious problem, but does anyone have
an idea what the problem is. Intermittently bad relay is my best guess.

Cheers