Engine & Props.

For a type certificated aircraft the only props that are allowed will be listed on the Type Certificate Data sheet.

Maybe the question was how does one choose the right prop for a given engine.

My guess would be that prop manufacturers have graphs of how much torque a prop can absorb at a given RPM.

And, RPM x torque = horsepower.

So e.g. a given prop might absorb 250HP at 2575 RPM, and your engine delivers say 250HP at 2575 RPM, so that combination should work.

That same prop might absorb 350HP at 2900 RPM. The power absorbed by a prop is IIRC proportional to the third power of the RPM, so a small RPM increase gives you a lot. It also gives you a lot more noise... but if you can get that (or some other) engine to deliver 350HP at 2900 RPM, there is another combination.

The more blades, the more power a prop can absorb at a given RPM. On GA piston engines, more than 2 are rarely needed, but 3 or even 4 looks a lot better

There are also secondary factors. An engine delivers its torque in lumps, as the combustion events take place. A 6 cylinder engine chucks out 3 "kicks" per revolution. This can play games with the natural resonance of the prop, so some props may not be suitable for a given engine.

Then there is ground clearance...

For fixed props: The most important factors are given a constant 75%/100% (cruise/take off) power : thrust in a specific speed range(cruise), static thrust (take off), diameter.

There are many more considerations: a 2 or 4 blade prop harmonizes better with a 4 cylinder, a 2 or 3 blade prop with a 6cyl engine.

a 2 or 4 blade prop harmonizes better with a 4 cylinder, a 2 or 3 blade prop with a 6cyl engine.

Why?

If that was the question, I could relate how my ground-adjustable prop was set/adjusted for it Rotax 912 engine. Basically: tie the plane to something really solid, and check the RPM at WOT. Adjust prop so that blades are at the same angle and the rpm is high enough. Rotax recommends 5200 rpm but mine was set for 4600, giving better cruise performance i.e. lower fuel burn, at the cost of needing more runway for take-off.

In addition to the correctly commented approvals required for certified aircraft, there are some additional factors. It is necessary that there is an acceptable vibration relationship between the prop and the engine, which is covered during the approval process. Further to that, there is a need for flight testing of the compatibility of the prop to the airframe, which is also part of the approval process.

While flight testing a small diameter four blade wooden propeller for approval on a Bellanca Scout, I learned an interesting, and terrifying lesson. I did all of the required flight testing from the front seat, which was all seemingly fine. For the glider tow operations, I moved into the back seat, so the qualified tow pilot could fly. Sitting in the back seat, after a few minutes of flight, my butt was numb from an odd vibration. I started considering this, and looking around. To my horror, I could see the leading edges of both H stabs rapidly vibrating up and down about an inch. I told the tow pilot to dump the glider ASAP, and get us back on the ground, with the least time and least airspeed possible.

We got back safely. I specified a structural examination of the rear fuselage, and did not approve the prop. Later, with a different one, we did succeed.

In addition to vibration characteristics, You also have to evaluate the potential for overspeed of the prop/engine during a dive, and assuring that the propeller pitch is such that it optimizes the engine power at full power (not too coarse)

For constant speed props, you must also optimize that relationship of the blade twist to the climb or cruise speed as required. That's quiet a science.

lastly, there is the relationship that the prop pitch can have to engine cooling.

I was once sent to ferry back a damaged Piper Tomahawk (my first time flying one). They had installed a replacement prop. It was the right prop model, but the wrong pitch. Static run up was 1900 RPM (I would have expected more in the 2300 to 2400 range). The takeoff took forever, and cruise RPM was about 2200, but it had a really good cruise speed! No harm done. as long as I had 4000 foot runways all the way home! The mistake was corrected very quickly!

Well, it seems we are not to know if the original question has more or less been answered. o/p would IMHO be polite to keep us posted.