Diophantine equations involving factorials

Horst Alzer; Florian Luca

Mathematica Bohemica (2017)

  • Volume: 142, Issue: 2, page 181-184
  • ISSN: 0862-7959

Abstract

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We study the Diophantine equations ( k ! ) n - k n = ( n ! ) k - n k and ( k ! ) n + k n = ( n ! ) k + n k , where k and n are positive integers. We show that the first one holds if and only if k = n or ( k , n ) = ( 1 , 2 ) , ( 2 , 1 ) and that the second one holds if and only if k = n .

How to cite

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Alzer, Horst, and Luca, Florian. "Diophantine equations involving factorials." Mathematica Bohemica 142.2 (2017): 181-184. <http://eudml.org/doc/288107>.

@article{Alzer2017,
abstract = {We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only if $k=n$.},
author = {Alzer, Horst, Luca, Florian},
journal = {Mathematica Bohemica},
keywords = {Diophantine equation; factorial},
language = {eng},
number = {2},
pages = {181-184},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Diophantine equations involving factorials},
url = {http://eudml.org/doc/288107},
volume = {142},
year = {2017},
}

TY - JOUR
AU - Alzer, Horst
AU - Luca, Florian
TI - Diophantine equations involving factorials
JO - Mathematica Bohemica
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 142
IS - 2
SP - 181
EP - 184
AB - We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only if $k=n$.
LA - eng
KW - Diophantine equation; factorial
UR - http://eudml.org/doc/288107
ER -

References

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  1. Andreescu, T., Andrica, D., Cucurezeanu, I., 10.1007/978-0-8176-4549-6, Birkhäuser, Basel (2010). (2010) Zbl1226.11001MR2723590DOI10.1007/978-0-8176-4549-6
  2. Bashmakova, I. G., Diophantus and Diophantine Equations, The Dolciani Mathematical Expositions 20. The Mathematical Association of America, Washington (1997). (1997) Zbl0883.11001MR1483067
  3. Carnal, H., 10.4171/EM/203, Elem. Math. 67 (2012), 151-154. (2012) Zbl1247.97035DOI10.4171/EM/203
  4. Luca, F., The Diophantine equation R ( x ) = n ! and a result of M. Overholt, Glas. Mat. (3) 37 (2002), 269-273. (2002) Zbl1085.11023MR1951531
  5. Luca, F., 10.3336/gm.48.1.03, Glas. Mat. (3) 48 (2013), 31-48. (2013) Zbl06201413MR3064240DOI10.3336/gm.48.1.03
  6. Sándor, J., On some Diophantine equations involving the factorial of a number, Seminar Arghiriade. Univ. Timişoara 21 (1989), 4 pages. (1989) Zbl0759.11011MR1124179

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