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

c³-c=b²+4

(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

c³-c-(b²+4)=0

since the solution (using Cardano’s formula) involves

$\sqrt[3]{\frac{b^2+4}{2}\pm\sqrt{\frac{(b^2+4)^2}{4}-\frac{1}{27}}}$

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).)