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### Beginning to Count the Number of Equivalent Resistances

#### Abstract

Given a bunch of equal resistors, what are the values of the equivalent resistances which one can obtain? Is there a formula? In this note, we address such questions analytically and establish strict lower and upper bounds for the order of the set of equivalent resistances using series and parallel connections. The approach of the Farey sequences is shown to be applicable to resistor networks involving bridge and non-planar circuits. For equal resistors, the order of the set is bounded from below by and from above by .

#### Keywords

Bridge, Fibonacci Numbers, Haros-Farey Sequences, Non-Planar Circuits, Parallel, Resistor Networks, Series.

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