Power is Work (or energy transfer of some kind) per unit time (let's call it rotations per minute). This can be written as:

Horsepower = Torque X RPM.

with a little algebra we get:

Horsepower/Torque = RPM

So you can see if you increase HP without increasing Torque, RPMs have to increase. Conversely, if you increase Torque without increasing HP, RPMs have to decrease. So if you increase BOTH Torque and HP, what the RPMS do depend on if you increase the HP in the numerator (RPMs increase) or the Torque in the denominator (RPMs decrease).

Hope this helps!

 

while the formula works out a cause effect, sometimes does not reflect real world as there are other variables.
while you work out the engine with the above formula it does change with gearing and more important btu potential of the fuel burn.
i will use a generator to illustrate a point:
you have a diesel unit driving a 1200 rpm generator with 100 amp load, the injector rack maintains rpm at said load, now if you add 100 more amps, the rpm stays the same but the fuel rack must open up for more fuel to burn. now if you increase load till the btu content can not supply the power needed, the generator slows down and trips on under hz or under voltage.
if you are tooling down the road and go into an incline, you must open the throttle to maintain speed through "X" gearing, but if you downshift, a rise in ratio increases torque but you need to burn "Y" fuel as rpm goes up. now suppose your engine produces "Z" power over the other it can climb the incline without a downshift but once again it is a function of fuel burn at "X" gearing.
was it newton that said for every action there is an equal and opposite reaction. so it comes down to balance of power (excluding parasitic loss).

 

Not sure what the responses are saying, but to answer your question— rpm required for a given speed is strictly dependent on gearing. Power has nothing to do with that fact.