Geodesic
On $x^n + y^n = \lowercase{n!} z^n$
Communications in Mathematics, Tome 26 (2018) no. 1, pp. 11-14 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\gcd (x, y, z)=1$.
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 $\gcd (x, y, z)=1$.
Classification : 11D41, 11D72
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. 
@article{COMIM_2018_26_1_a1,
     author = {Jena, Susil Kumar},
     title = {On $x^n + y^n = \lowercase{n!} z^n$},
     journal = {Communications in Mathematics},
     pages = {11--14},
     year = {2018},
     volume = {26},
     number = {1},
     mrnumber = {3827140},
     zbl = {06996470},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2018_26_1_a1/}
}
TY  - JOUR
AU  - Jena, Susil Kumar
TI  - On $x^n + y^n = \lowercase{n!} z^n$
JO  - Communications in Mathematics
PY  - 2018
SP  - 11
EP  - 14
VL  - 26
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/COMIM_2018_26_1_a1/
LA  - en
ID  - COMIM_2018_26_1_a1
ER  - 
%0 Journal Article
%A Jena, Susil Kumar
%T On $x^n + y^n = \lowercase{n!} z^n$
%J Communications in Mathematics
%D 2018
%P 11-14
%V 26
%N 1
%U http://geodesic.mathdoc.fr/item/COMIM_2018_26_1_a1/
%G en
%F COMIM_2018_26_1_a1
Jena, Susil Kumar. On $x^n + y^n = \lowercase{n!} z^n$. Communications in Mathematics, Tome 26 (2018) no. 1, pp. 11-14. http://geodesic.mathdoc.fr/item/COMIM_2018_26_1_a1/

[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

[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 | Zbl

[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 | MR