How can we prove no Pythagorean Triple can be written as \(({x^{n} ,y^{n},z^{n}})\) for integer \(x,y,z,n\) and \(n>1\) without using Fermat's Last Theorem?

Note that if we could prove this, we would have proven half of Fermat's Last Theorem since \(x^{n}+y^{n}=z^{n}\) can be written as \((x^{m})^{2}+(y^{m})^{2}=(z^{m})^{2}\) for even \(n\) and \(m>1\).

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