| Title:
|
On $x^n + y^n = \lowercase{n!} z^n$ (English) |
| Author:
|
Jena, Susil Kumar |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
26 |
| Issue:
|
1 |
| Year:
|
2018 |
| Pages:
|
11-14 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $\gcd (x, y, z)=1$. (English) |
| Keyword:
|
Diophantine equation $x^{n} + y^{n} = \lowercase {n!} z^{n}$ |
| Keyword:
|
Diophantine equation $x^{3} + y^{3} = \lowercase {3!} z^{3}$ |
| Keyword:
|
unsolved problems |
| Keyword:
|
number theory. |
| MSC:
|
11D41 |
| MSC:
|
11D72 |
| idZBL:
|
Zbl 06996470 |
| idMR:
|
MR3827140 |
| . |
| Date available:
|
2018-11-06T16:17:28Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147454 |
| . |
| Reference:
|
[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 |
| Reference:
|
[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, |
| Reference:
|
[3] Guy, R. K.: Unsolved Problems in Number Theory.2004, Springer Science+Business Media, Inc., New York, Third Edition.. Zbl 1058.11001, MR 2076335 |
| Reference:
|
[4] Ribet, K.: On modular representations of Gal($\overline {\mathbb Q}\setminus \mathbb {Q}$) arising from modular forms.Invent. Math., 100, 1990, 431-476, MR 1047143, 10.1007/BF01231195 |
| . |
