English

On the Diophantine equation $f(x)=2f(y)$

Number Theory 2018-11-06 v2
Authors: Sanjay Bhatter , Richa Sharma
View on arXiv ↗ PDF ↗

Abstract

Let f(x)=x2(x2−1)(x2−2)(x2−3).f(x)=x^{2}(x^{2}-1)(x^{2}-2)(x^{2}-3). We prove that the Diophantine equation f(x)=2f(y) f(x)=2f(y) has no solutions in positive integers xx and yy, except (x,y)=(1,1)(x, y)=(1, 1).

Keywords

diophantine equation positive integers polynomial equation integer solutions number theory

Cite

@article{arxiv.1810.06817,
  title  = {On the Diophantine equation $f(x)=2f(y)$},
  author = {Sanjay Bhatter and Richa Sharma},
  journal= {arXiv preprint arXiv:1810.06817},
  year   = {2018}
}

Comments

I want to improve the results

Related papers

View all related →
Number Theory · Mathematics
The diophantine equation $\left(2^{k}-1\right)\left(3^{k}-1\right)=x^{n}$
Bo He, Chang Liu
2025-07-29
Number Theory · Mathematics
Diophantine approximation and the equation (a^2 c x^k - 1)(b^2 c y^k - 1) = (abc z^k - 1)^2
E. G. Goedhart, H. G. Grundman
2014-11-10
Number Theory · Mathematics
On the Diophantine equation N X^2 + 2^L 3^M = Y^N
Eva G. Goedhart, Helen G. Grundman
2014-04-18
Number Theory · Mathematics
On the Diophantine equation x^2+2^a.3^b.11^c=y^n
Ismail Naci Cangul, Musa Demirci, Ilker Inam, Florian Luca +1
2012-01-04
Number Theory · Mathematics
Problems Regarding a Pair of Diophantine Equations
Hung Viet Chu, Steven J. Miller, Garrett Tresch
2025-12-16
Number Theory · Mathematics
A note on the Diophantine equations $x^{2}\pm5^{\alpha}\cdot p^{n}=y^{n}$
Gökhan Soydan
2017-06-13
Number Theory · Mathematics
On the exponential Diophantine equation $(n-1)^{x}+(n+2)^{y}=n^{z}$
Hairong Bai, Elif Kızıldere, Gökhan Soydan, Pingzhi Yuan
2020-03-31
Number Theory · Mathematics
The diophantine equation $(2^{k}-1)(b^{k}-1)=y^{q}$
Chang Liu, Bo He
2026-04-09
Number Theory · Mathematics
A Pair of Diophantine Equations and Fibonacci-Like Sequences
Hung Viet Chu, Rishabh Gulecha, Sicheng Guo, Nathanael Johnson +2
2025-09-11
Number Theory · Mathematics
On the Exponential Diophantine Equation $(a^n-1)(b^n-1)=x^2$
Armand Noubissie, Alain Togbe, Zhongfeng Zhang
2025-04-22
Number Theory · Mathematics
A Note On the Exponential Diophantine Equation (a^n-1)(b^n-1)=x^2
Refik Keskin
2018-01-16
Number Theory · Mathematics
On some conjectures of exponential Diophantine equations
Hairong Bai
2021-01-01
Number Theory · Mathematics
On the Diophantine equation $x^2+q^{2m}=2y^p$
Szabolcs Tengely
2015-06-26
General Mathematics · Mathematics
A Method to Solve the Diophantine Equation $ax^2-by^2+c=0$
Florentin Smarandache
2007-05-23
Number Theory · Mathematics
Diophantine equations involving Euler function
Hairong Bai
2020-01-24
Number Theory · Mathematics
On the Exponential Diophantine Equation $(a^2-2)(b^2-2)=x^2$
Zafer Şiar, Refik Keskin
2018-01-16
Number Theory · Mathematics
The Diophantine equation $aX^{4} - bY^{2} = 1$
Shabnam Akhtari
2009-03-11
Number Theory · Mathematics
A generalization of the Ramanujan-Nagell equation
Tomohiro Yamada
2017-12-07
Number Theory · Mathematics
Resolution of the equation $(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$
Kálmán Liptai, László Németh, Gökhan Soydan, László Szalay
2021-04-01
Number Theory · Mathematics
On the Diophantine equations f (x) = g(y)
S. Subburam, J. Tanti
2019-02-12
Number Theory · Mathematics
Diophantine equations involving Euler's totient function
Yong-Gao Chen, Hao Tian
2022-01-27
Number Theory · Mathematics
An Exponential Diophantine equation $x^2+3^{\alpha} 113^{\beta}=y^{\mathfrak{n}}$
S. Muthuvel, R. Venkatraman
2024-05-20
Number Theory · Mathematics
A Pair of Diophantine Equations Involving the Fibonacci Numbers
Xuyuan Chen, Hung Viet Chu, Fadhlannafis K. Kesumajana, Dongho Kim +4
2024-09-06
Number Theory · Mathematics
On the Diophantine equation $B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$
Kisan Bhoi, Prasanta Kumar Ray
2023-06-22
Number Theory · Mathematics
On the Diophantine equation cy^l=(x^p-1)/(x-1)
Mohammad Sadek
2013-02-26