Which nut first?

sort of. The nuts force the prop tightly onto the taper after that they are just back up. Props properly installed require a puller to remove and some have been found with nuts missing but the prop still in place. Personally I favor trouping the big nut against the pro then removing it and adding the small nut and the big nut in that order with proper cotter pin or wire.

A common cause of loos or vibrating or loose props is a key that is too far up the ramp when the prop is installed. That prevents proper seating of the prop on the shaft.
 
There are 3 things to consider in all of this torque, friction and yield strength.

Shear strength has no bearing.

When torquing a traditional nut/bolt the goal is to achieve a maximum clamping force typically 60-70% of yield strength. This allows the bolt which is elastic in nature to stretch thereby get narrower. This allows the bolt to return to its normal state when the load is removed. Over torquing a bolt beyond that simply stretchs the bolt to the point that it is no longer elastic and does not return to it's normal state.

The application of torque required to achieve that desired result is a function of material and friction. Dry mating surfaces require more torque and run a greater risk of galling. Even using water as a lubricant while not recommended reduces the amount of torque required.

That said, in the case of a prop on a tapered shaft one could argue that last thing you would want to do is torque a nut so as to elongated the shaft. Props are held in place more as a function of friction and a key way to prevent rotation of the prop in relation to the shaft. The nuts are simply a safety device. As such once the prop is seated on the shaft and the smaller nut is set against the shoulder of the prop/shaft the only elongation of the shaft would be minimal and happen when the larger nut is torqued against the smaller nut. The stretch would be between the nuts.
 
I always seated the prop with the large nut first, then backed it off and placed the jam nut on and snugged it up, then placed the larger nut on and snugged it up, before placing slotted cap and cotter pin on. Always worked for me with zero problems.
 
I agree with Rob, L Keith & Cam above. Seat 1st with the big nut to torque using more threads , then switch to leave big nut on last.
And mine are torqued to about 50ft-lbs with 19"D props on 1 1/4" Aquamet 22(?) tapered-end shafts.

If the prop were to try to back off the taper-grip with hard reversing, the thin nut next to the hub would tend to jam harder locking against the big torque nut that holds everything in place with force on the forward side of more shaft threads. That's good.

If you leave the thick nut against the prop and then tighten the thin nut against it, the stronger thick nut becomes unloaded from the forward side of the shaft threads, opposite of what is desired .
 
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