The diophantine equation x 2 + 2 a · 17 b = y n

Su Gou; Tingting Wang

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 645-654
  • ISSN: 0011-4642

Abstract

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Let ℤ , ℕ be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, there have been many papers concerned with solutions ( x , y , n , a , b ) of the equation x 2 + 2 a p b = y n , x , y , n ∈ ℕ , gcd ( x , y ) = 1 , n ≥ 3 , a , b ∈ ℤ , a ≥ 0 , b ≥ 0 . And all solutions of it have been determined for the cases p = 3 , p = 5 , p = 11 and p = 13 . In this paper, we mainly concentrate on the case p = 3 , and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions ( x , y , n , a , b ) of the equation x 2 + 2 a · 17 b = y n , x , y , n ∈ ℕ , gcd ( x , y ) = 1 , n ≥ 3 , a , b ∈ ℤ , a ≥ 0 , b ≥ 0 , are determined.

How to cite

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Gou, Su, and Wang, Tingting. "The diophantine equation $x^2+2^a\cdot 17^b=y^n$." Czechoslovak Mathematical Journal 62.3 (2012): 645-654. <http://eudml.org/doc/247237>.

@article{Gou2012,
abstract = {Let $\mathbb \{Z\}$, $ \mathbb \{N\}$ be the sets of all integers and positive integers, respectively. Let $p$ be a fixed odd prime. Recently, there have been many papers concerned with solutions $(x, y, n, a, b)$ of the equation $ x^2+2^ap^b=y^n$, $x, y, n\in \mathbb \{N\}$, $\gcd (x, y)=1$, $n\ge 3$, $a, b\in \mathbb \{Z\}$, $a\ge 0$, $b\ge 0. $ And all solutions of it have been determined for the cases $p=3$, $p=5$, $p=11$ and $p=13$. In this paper, we mainly concentrate on the case $p=3$, and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions $(x, y, n, a, b)$ of the equation $x^2+2^a\cdot 17^b=y^n$, $x, y, n\in \mathbb \{N\}$, $\gcd (x, y)=1$, $n\ge 3$, $a, b\in \mathbb \{Z\}$, $ a\ge 0$, $ b\ge 0$, are determined.},
author = {Gou, Su, Wang, Tingting},
journal = {Czechoslovak Mathematical Journal},
keywords = {exponential diophantine equation; modular approach; arithmetic properties of Lucas numbers; exponential diophantine equation; modular approach; Lucas number},
language = {eng},
number = {3},
pages = {645-654},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The diophantine equation $x^2+2^a\cdot 17^b=y^n$},
url = {http://eudml.org/doc/247237},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Gou, Su
AU - Wang, Tingting
TI - The diophantine equation $x^2+2^a\cdot 17^b=y^n$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 645
EP - 654
AB - Let $\mathbb {Z}$, $ \mathbb {N}$ be the sets of all integers and positive integers, respectively. Let $p$ be a fixed odd prime. Recently, there have been many papers concerned with solutions $(x, y, n, a, b)$ of the equation $ x^2+2^ap^b=y^n$, $x, y, n\in \mathbb {N}$, $\gcd (x, y)=1$, $n\ge 3$, $a, b\in \mathbb {Z}$, $a\ge 0$, $b\ge 0. $ And all solutions of it have been determined for the cases $p=3$, $p=5$, $p=11$ and $p=13$. In this paper, we mainly concentrate on the case $p=3$, and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions $(x, y, n, a, b)$ of the equation $x^2+2^a\cdot 17^b=y^n$, $x, y, n\in \mathbb {N}$, $\gcd (x, y)=1$, $n\ge 3$, $a, b\in \mathbb {Z}$, $ a\ge 0$, $ b\ge 0$, are determined.
LA - eng
KW - exponential diophantine equation; modular approach; arithmetic properties of Lucas numbers; exponential diophantine equation; modular approach; Lucas number
UR - http://eudml.org/doc/247237
ER -

References

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