On $x^n+y^n=n!z^n$

Jena, Susil Kumar. "On $x^n + y^n = n! z^n$." Communications in Mathematics 26.1 (2018): 11-14. <http://eudml.org/doc/294153>.

@article{Jena2018,
abstract = {In p. 219 of R.K. Guy’s Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^\{n\} + y^\{n\} = n! z^\{n\}$ 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 $\gcd (x, y, z)=1$.},
author = {Jena, Susil Kumar},
journal = {Communications in Mathematics},
keywords = {Diophantine equation $x^\{n\} + y^\{n\} = n! z^\{n\}$; Diophantine equation $x^\{3\} + y^\{3\} = 3! z^\{3\}$; unsolved problems; number theory},
language = {eng},
number = {1},
pages = {11-14},
publisher = {University of Ostrava},
title = {On $x^n + y^n = n! z^n$},
url = {http://eudml.org/doc/294153},
volume = {26},
year = {2018},
}

TY - JOUR
AU - Jena, Susil Kumar
TI - On $x^n + y^n = n! z^n$
JO - Communications in Mathematics
PY - 2018
PB - University of Ostrava
VL - 26
IS - 1
SP - 11
EP - 14
AB - In p. 219 of R.K. Guy’s Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^{n} + y^{n} = n! z^{n}$ 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 $\gcd (x, y, z)=1$.
LA - eng
KW - Diophantine equation $x^{n} + y^{n} = n! z^{n}$; Diophantine equation $x^{3} + y^{3} = 3! z^{3}$; unsolved problems; number theory
UR - http://eudml.org/doc/294153
ER -