Problem

Find all integer solutions to

$$ 19x^3-84y^2=1984. $$

Solution - by Rishith (India)

We consider the same equation modulo $7$:

$$ 5x^3 \equiv 3 \mod 7. $$

To solve for this, we compute $5x^3 \mod 7$ from $x = 0, \dots 6$:

No value possible for $x$ satisfies $5x^3 = 3 \mod 7$, hence there are no integer solutions to the above equation.

Thank you!

Huge thanks to Hagan Chan (Hong Kong), Anisur Rahaman (India), Valter (Sweden), Jake Sun (Boston), Uddhav Venkatesh (London), Naitik (Switzerland), Ammar Ratnani (Stanford), Valter (Sweden), and Rishith (India) for submitting solutions for this challenge problem!