Archive for cubic polynomial

Diophantine riddle

Posted in Books, Kids, R with tags , , , , , , , on February 27, 2023 by xi'an

The weekly riddle from The Riddler is to find solutions to the Diophantine equation


(when b and c are positive integers). First, forget about ChatGPT since it states this is a Pell equation. With a wrong argument. Second, when running a basic R code, using as.double to handle larger integers, the only solution less than 10⁶ this code returned was

[1]    999799 999700015

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


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 is simpler: the lhs is a multiple of 3, while the rhs cannot be, as shown by looking at b(3).)

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