On the Diophantine equation x 2 - k x y + y 2 - 2 n = 0

Refik Keskin; Zafer Şiar; Olcay Karaatlı

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 3, page 783-797
  • ISSN: 0011-4642

Abstract

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In this study, we determine when the Diophantine equation x 2 - k x y + y 2 - 2 n = 0 has an infinite number of positive integer solutions x and y for 0 ≤ n ≤ 10 . Moreover, we give all positive integer solutions of the same equation for 0 ≤ n ≤ 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x 2 - k x y + y 2 - 2 n = 0 .

How to cite

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Keskin, Refik, Şiar, Zafer, and Karaatlı, Olcay. "On the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$." Czechoslovak Mathematical Journal 63.3 (2013): 783-797. <http://eudml.org/doc/260652>.

@article{Keskin2013,
abstract = {In this study, we determine when the Diophantine equation $x^\{2\}-kxy+y^\{2\}-2^\{n\}=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\le n\le 10.$ Moreover, we give all positive integer solutions of the same equation for $0\le n\le 10$ in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation $x^\{2\}-kxy+y^\{2\}-2^\{n\}=0$.},
author = {Keskin, Refik, Şiar, Zafer, Karaatlı, Olcay},
journal = {Czechoslovak Mathematical Journal},
keywords = {Diophantine equation; Pell equation; generalized Fibonacci number; generalized Lucas number; quadratic Diophantine equation; Pell equation; generalized Fibonacci number; generalized Lucas number},
language = {eng},
number = {3},
pages = {783-797},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Diophantine equation $x^\{2\}-kxy+y^\{2\}-2^\{n\}=0$},
url = {http://eudml.org/doc/260652},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Keskin, Refik
AU - Şiar, Zafer
AU - Karaatlı, Olcay
TI - On the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 783
EP - 797
AB - In this study, we determine when the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\le n\le 10.$ Moreover, we give all positive integer solutions of the same equation for $0\le n\le 10$ in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$.
LA - eng
KW - Diophantine equation; Pell equation; generalized Fibonacci number; generalized Lucas number; quadratic Diophantine equation; Pell equation; generalized Fibonacci number; generalized Lucas number
UR - http://eudml.org/doc/260652
ER -

References

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