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Keywords:
Diophantine equation $x^{n} + y^{n} = \lowercase {n!} z^{n}$; Diophantine equation $x^{3} + y^{3} = \lowercase {3!} z^{3}$; unsolved problems; number theory.
Diophantine equation $x^{n} + y^{n} = \lowercase {n!} z^{n}$; Diophantine equation $x^{3} + y^{3} = \lowercase {3!} z^{3}$; unsolved problems; number theory.
Summary:
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 $\gcd (x, y, z)=1$.
References:
[1] Elkies, N. D.: Wiles minus epsilon implies Fermat. Elliptic Curves, Modular Forms & Fermat's Last Theorem, 1995, 38-40, Ser. Number Theory, I, Internat. Press, Cambridge MA.. MR 1363494
[2] Erdös, P., Obláth, R.: Über diophantische Gleichungen der form $n! = x^p \pm y^p$ and $n! \pm m! = x^p$. Acta Litt. Sci. Szeged, 8, 1937, 241-255,
[3] Guy, R. K.: Unsolved Problems in Number Theory. 2004, Springer Science+Business Media, Inc., New York, Third Edition.. MR 2076335 | Zbl 1058.11001
[4] Ribet, K.: On modular representations of Gal($\overline {\mathbb Q}\setminus \mathbb {Q}$) arising from modular forms. Invent. Math., 100, 1990, 431-476, DOI 10.1007/BF01231195 | MR 1047143
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