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NUMBER THEORETIC TECHNIQUES IN THE SETS OF EQUAL RESISTOR NETWORKS
Keywords:
resistor network, number theory, Fibonacci numbers, Haros-Farey sequence, series, parallel, bridge and non-planar circuitsDOI:
https://doi.org/10.17654/0972087125037Abstract
A variety of sets of equivalent resistances can be constructed by combining equal resistors in different configurations using series, parallel or bridge connections. The orders of such sets are traditionally obtained manually for small numbers and computationally, when the numbers are not small. Due to available computer memory, it has not been possible to analyze these sets beyond the case of 30 equal resistors. In this article, we analytically derive a strict lower bound and a strict upper bound for the orders of these sets using inequalities, integer sequences and certain number theoretic techniques. The strict lower bound is obtained using combinatorial arguments and the resulting inequalities. The strict upper bound is obtained by using techniques from number theory, which make use of the Haros-Farey sequence with Fibonacci numbers as their argument. The lower and upper bounds thus obtained hold true for any number of equal resistors as long as they are combined in series and parallel combinations. The various sequences arising in this study are presented in detail with references to the “The On-Line Encyclopedia of Integer Sequences”.
Received: April 13, 2025
Accepted: June 25, 2025
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