Gas turbine engines have maximum speed and power limits based on pressures, temperatures and mechanical loads on the parts of the turbine spinning at ludicrous speed.
So…what parameters determine the minimum RPM for a gas turbine? When that 747 is waiting for takeoff, all four engines are “idling,” putting out the least thrust (and consuming the least fuel) possible. What exactly happens if they try to spin the engines at an even lower RPM?
The need to spin the turbines fast enough to get the compressor and fan to push enough air into the combustor to maintain combustion. Get too little air into the combustor, and therefore back fuel flow down too far in an effort to maintain the mixture, and you’ll flame out. Typically that min RPM is about half of the maximum RPM, which is high compared to the min RPM on a piston engine, because most of the energy put out by the combustor actually goes into just running the compressor, not into pushing the airplane.
As an example of what ElvisLives is saying, the turboprop I fly burns about 550 lbs of fuel per hour when sitting on the ground with the engines idling, but only 1050 lbs per hour when flying at 250 knots.
It should be obvious that the minimum RPM for any jet engine is zero.
The minimum sustainable RPM is much higher. I can’t find the actual rpm numbers for our engines but the published minimum speed of the high pressure compressor/turbine spool is 66% of the rated maximum. If the engine speed gets much lower than that the ignitors kick in to prevent a flame out.
From my Boeing 737 book, the CFM56 engines on the 737 (and some of its Airbus competitors), the two compressor sections max RPM is 5175 RPM for the N1 (low pressure) compressor, and 14460 RPM for the N2 (high pressure) compressor.
Usually the CFM56 will idle around 22% to 27%, give or take. When starting, fuel is added at 25% N1. Taxi thrust is around 35% IIRC, and takeoff thrust can be from 80% to 100%, depending on how the aircraft is set up, runway length, weather, etc. You can take the math from there for the actual RPMs.
I don’t know how close those numbers are to other aircraft.