Solution Overview
Problem 1213 was solved by David Broadhurst, Joseph DeVincentis, Russ Gordon, Richard Boardman, John Snyder, and Franz Pichler.
It was first posed by Dick Hess in the Pi Mu Epsilon Journal; the notes in this PDF follow the solution of William Pierce (Pi Mu Epsilon Journal, Problem 971, 11:3 Fall 2000, 159-160).
They start with an elementary solution to the problem, and then discuss a more sophisticated approach using Chebyshev polynomials (David Broadhurst, Russ Gordon), which could lead to a proof of minimality of the smallest solution. There is also an approach using Gaussian integers (Walter Taylor). Those approaches also allow an easy algorithm to solve the problem for any other ratio in place of 5:7.
Download the PDF.
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© Copyright 2015 Stan Wagon. Reproduced with permission.