Solve the equation over $\textbf{Z}$ :
$x^3$ - 3$y$ = 2
The only way I solve this problem was using the Fermat Theorem. Is there any chance to solve it without using the theorem? And the proof to be explained for the primary/secondary school children?
I don't know any theorems about primary school children. Take any integer $x = 3k-1$ for any integer $k,$ then $x^3 = 27k^3 - 27 k^2 + 9 - 1.$ So $x^3 - 2 = 27k^3 - 27 k^2 + 9 - 3.$ And $$ \frac{x^3 - 2}{3} = 9k^3 - 9 k^2 + 3 - 1.$$ So, take $$ x=3k-1, \; \; \; y = 9k^3 - 9 k^2 + 3 - 1. $$