October 3, 2023

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[This article was first published on R – Xi’an’s Og, and kindly contributed to R-bloggers], (You can report content on this page here) Want to share your content on R-Bloggers? Click here if you have a blog, or click here if you don’t have one.

The Riddler’s weekly puzzle is to find the solution to the Diophantine equation.

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(when b and c are positive integers). First of all, forget about ChatGPT as it states that it is a Pell equation. by wrong reasoning. Second, when running the original R code, using as.double to handle large integers, the only solution less than 10⁶ this code returned was

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[1] 999799 999700015

The first column is c and the second is b. But this is not the correct solution!, as confirmed by Mathematica, which states that there is no integer solution. This makes sense when looking at the unique real solution (in C) of the cube

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Since the solution (using Cardano’s formula) involves

leaving the inverse of 27 as the only non-integer term in the expression when b is even … (the exact solution that this Diophantine equation has no solution: lhs is a multiple of 3, while rhs cannot be, as is shown by looking at B(3).)


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