The equation $X^{n} + Y^{n} = Z^{n}$ , where $ n \geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are intergers.

Question

The equation $X^{n} + Y^{n} = Z^{n}$ , where $n \geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are integers.

My solution:

False. Because if we let $X=0 ,Y=0$, then $0^{3}+0^{3}=0^{3}$ . Thus there is a solution for the equation $X^{n} + Y^{n} = Z^{n}$, where $X,Y,Z$ are integers.

Can anyone please give feedback on my answer and state whether it is correct on not.

Answer 1

You are correct. Disproof by counterexample is valid.

For a bit stronger and correct variation of this, see Fermat's Last theorem.