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Is there an equation which allow us to perfectly match a helix propeller and an engine? I'm looking for an equation that allow me to correctly dimension an helix propeller for my engine. I mean that allow me to maximize efficiency and thrust, calculate the optimal number of blades, the optimal number of rpm, and the maximum rpm of my helix propeller given a certain engine power, etc.

Should I ask this question on the physics division of this website?

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Did you try Mark Drela's XROTOR? I have difficulty with your term "helix", but that software should do what you ask if your blade loading is not too high.

From the linked page: XROTOR is an interactive program for the design and analysis of ducted and free-tip propellers and windmills. It consists of a collection of menu-driven routines which perform various useful functions such as:

  • Design of minimum induced loss rotor (propeller or windmill)
  • Prompted input of an arbitrary rotor geometry
  • Interactive modification of a rotor geometry
  • Twist optimization of an arbitrary rotor for minimum induced loss
  • Analysis of a rotor with a wealth of choices of operating parameters
  • Incoming slipstream effects (from an upstream propeller, viscous wake...)
  • Multi-point parameter display
  • Structural analysis and corrections for twist under load
  • Acoustic analysis with dB noise footprint predictions
  • Interpolation of geometry to radii of interest
  • Plotting of geometry, aerodynamic parameters, and performance maps
  • ...
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