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}}}](https://s0.wp.com/latex.php?latex=%5Csqrt%5B3%5D%7B%5Cfrac%7Bb%5E2%2B4%7D%7B2%7D%5Cpm%5Csqrt%7B%5Cfrac%7B%28b%5E2%2B4%29%5E2%7D%7B4%7D-%5Cfrac%7B1%7D%7B27%7D%7D%7D&bg=000000&fg=B0B0B0&s=0&c=20201002)
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).)
Xi'an's Og 