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Diophantine equation $x^{n} + y^{n} = \lowercase {n!} z^{n}$; Diophantine equation $x^{3} + y^{3} = \lowercase {3!} z^{3}$; unsolved problems; number theory.
In p.~219 of R.K. Guy's \emph {Unsolved Problems in Number Theory}, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^{n} + y^{n} = \lowercase {n!} z^{n}$ has no integer solutions with $n\in \mathbb {N_{+}}$ and $n>2$. But, contrary to this expectation, we show that for $n = 3$, this equation has infinitely many primitive integer solutions, i.e.~the solutions satisfying the condition $\gcd (x, y, z)=1$.
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