You lost me at "invoking some number theory you can get down to 575 resistors."
If you want to get 1.337 milliOhms from as few 1% 1 Ohm resistors as possible, you can't do better than putting them all in parallel.
With 3 significant digits, 748 resistors in parallel would be within the tolerance of the resistors.
Do you know of some magical way of wiring resistors that is not a series-parallel circuit?
- A resistor network is made up of series-parallel circuits
- Series: Rtotal=R1+R2+R3+...+Rn
- Parallel: Rtotal=(1)/((1/R1)+(1/R2)+(1/R3)+...+(1/Rn))
- Resistor networks can be broken into sub-networks
Our goal is to lower the resistance an arbitrary amount with as few resistors as possible.
Claim: The resistor network of the chosen resistance with the fewest resistors will have no resistors in series iff extra accuracy is not needed.
Proof by restatement of Claim from Premises.
- The smallest possible resistor network is a single resistor. (by 1 and 4)
- Resistor networks in series always raise the resistance. (by 2 and A)
- Resistor networks in parallel always lower the resistance. (by 3 and A)
- The simplest resistor network is formed with a single resistor. (by A)
- To lower the resistance, extra resistors must be added in parallel. (by 1, B and C)
- Recursively add a single resistor in parallel until desired resistance is overshot or met. (until resistance <= target resistance)
- At the end of recursion we have the simplest possible resistor network with the desired resistance. (By D, E and F)
- The simplest possible resistor network with the desired resistance has no resistors in series. (by F and G)
Edit3: Well, not quite: I introduced the Goal
at step E.