How to prove that there is no solution for this equation in natural numbers:
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Did you check the equation mod 2, 3, 5? There, you can analyse all cases, and maybe no case can satisfy the equation - in this case, it is impossible to satisfy your equation.
What if i take mod 3 ,how is that show that there is no solution ?
Mod 3 doesn't help, because the LHS is 1 (mod 3) and $\displaystyle k^3$ can also be congruent to 1 (mod 3). However, mod 9 gives a nice solution. Try using mod 9.
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