I almost thought that, but normally under the conditions described a bad
thermocouple would not allow the pilot to say on


When I try to turn the pilot adjustment knob to "ON"
the pilot light goes out even if when the pilot has been
burning for 20 min (hot). The knob in
question has three settings - 1. Off, 2. Pilot, 3. On.

Sounds exactly like the TC to me.