Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Priyanka Baruah; Anup Das; Azizul Hoque

Archivum Mathematicum (2024)

  • Volume: 060, Issue: 3, page 135-144
  • ISSN: 0044-8753

Abstract

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We consider the Lebesgue-Ramanujan-Nagell type equation x 2 + 5 a 13 b 17 c = 2 m y n , where a , b , c , m ≥ 0 , n ≥ 3 and x , y ≥ 1 are unknown integers with gcd ( x , y ) = 1 . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all S -integral points on a class of elliptic curves.

How to cite

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Baruah, Priyanka, Das, Anup, and Hoque, Azizul. "Complete solutions of a Lebesgue-Ramanujan-Nagell type equation." Archivum Mathematicum 060.3 (2024): 135-144. <http://eudml.org/doc/299460>.

@article{Baruah2024,
abstract = {We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a13^b17^c=2^m y^n$, where $a,b,c, m\ge 0, n \ge 3$ and $x, y\ge 1$ are unknown integers with $\gcd (x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves.},
author = {Baruah, Priyanka, Das, Anup, Hoque, Azizul},
journal = {Archivum Mathematicum},
keywords = {Diophantine equation; Lehmer sequence; elliptic curve; quartic curve; S-integral points},
language = {eng},
number = {3},
pages = {135-144},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Complete solutions of a Lebesgue-Ramanujan-Nagell type equation},
url = {http://eudml.org/doc/299460},
volume = {060},
year = {2024},
}

TY - JOUR
AU - Baruah, Priyanka
AU - Das, Anup
AU - Hoque, Azizul
TI - Complete solutions of a Lebesgue-Ramanujan-Nagell type equation
JO - Archivum Mathematicum
PY - 2024
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 060
IS - 3
SP - 135
EP - 144
AB - We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a13^b17^c=2^m y^n$, where $a,b,c, m\ge 0, n \ge 3$ and $x, y\ge 1$ are unknown integers with $\gcd (x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves.
LA - eng
KW - Diophantine equation; Lehmer sequence; elliptic curve; quartic curve; S-integral points
UR - http://eudml.org/doc/299460
ER -

References

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